diff --git a/compose-rust/android-kotlin/shared/ComposeRustHost.kt b/compose-rust/android-kotlin/shared/ComposeRustHost.kt
index 03c40c11..681dcea3 100644
--- a/compose-rust/android-kotlin/shared/ComposeRustHost.kt
+++ b/compose-rust/android-kotlin/shared/ComposeRustHost.kt
@@ -484,6 +484,7 @@ private fun ComposeRustTree(
         chainOf = { asset -> (host.table.assets.asset(asset) as? Asset.Font)?.chain },
         brushOf = { asset -> (host.table.assets.asset(asset) as? Asset.Brush)?.brush },
         windowBackdrop = asked && platformBacksWindowWithMaterial(),
+        overlay = host.table.overlayTokens,
     )
     // The animations the Host handed over are played here, on this side's frame clock, and
     // the Host is not called while they play unless one asked for events. The loop waits
diff --git a/compose-rust/android-kotlin/shared/DesignSystem.kt b/compose-rust/android-kotlin/shared/DesignSystem.kt
index 682b9586..f603e80f 100644
--- a/compose-rust/android-kotlin/shared/DesignSystem.kt
+++ b/compose-rust/android-kotlin/shared/DesignSystem.kt
@@ -22,6 +22,7 @@ import dev.darkpyonix.composerust.protocol.DesignTokens
 import androidx.compose.animation.core.FiniteAnimationSpec
 import dev.darkpyonix.composerust.protocol.MaterialRole
 import dev.darkpyonix.composerust.protocol.MotionRole
+import dev.darkpyonix.composerust.protocol.OverlayTokens
 import dev.darkpyonix.composerust.protocol.Paint
 import dev.darkpyonix.composerust.protocol.ShapeRole
 import dev.darkpyonix.composerust.protocol.SpaceRole
@@ -180,6 +181,11 @@ class ResolvedTheme(
      * given is enough to find the rest of the stack: see [chainFor].
      */
     val fontChains: List<FontChain> = emptyList(),
+
+    /**
+     * Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet.
+     */
+    val overlay: OverlayTokens? = null,
 ) {
     /**
      * The stack [family] is the first family of, or null where it is not one.
@@ -1638,6 +1644,8 @@ fun resolveTheme(
      * family lacks in the families after it.
      */
     chainOf: (Int) -> FontChain? = { null },
+    /** Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+    overlay: OverlayTokens? = null,
 ): ResolvedTheme {
     val system = when {
         theme == null -> adaptiveSystem(platform, DesignSystem.Material3)
@@ -1684,6 +1692,7 @@ fun resolveTheme(
         platform,
         application,
         chains,
+        overlay,
     )
 }
 
diff --git a/compose-rust/android-kotlin/shared/HostContainer.kt b/compose-rust/android-kotlin/shared/HostContainer.kt
index 9914be33..bafe5709 100644
--- a/compose-rust/android-kotlin/shared/HostContainer.kt
+++ b/compose-rust/android-kotlin/shared/HostContainer.kt
@@ -7,6 +7,7 @@ import dev.darkpyonix.composerust.protocol.TypeRole
 import dev.darkpyonix.composerust.ui.textStyle
 import androidx.compose.foundation.background
 import androidx.compose.foundation.border
+import androidx.compose.foundation.shape.RoundedCornerShape
 import androidx.compose.foundation.layout.Arrangement
 import androidx.compose.foundation.layout.Box
 import androidx.compose.foundation.layout.Column
@@ -45,12 +46,20 @@ import dev.darkpyonix.composerust.ui.resolvedShape
 import dev.darkpyonix.composerust.ui.weightOf
 import dev.darkpyonix.composerust.design.liftsOffThePage
 
+/** True when the Host set a shape modifier of its own, which replaces the default shape. */
+internal fun Node.setsOwnShape(): Boolean =
+    modifiers.any {
+        it is ProtocolModifier.Shape ||
+            it is ProtocolModifier.ShapeRole ||
+            it is ProtocolModifier.CornerEach
+    }
+
 /**
  * The corner this container is drawn with: the one the Host asked for if it asked, and the
  * design system's corner for this kind of container otherwise.
  */
 internal fun Node.containerShape(style: ContainerStyle, theme: ResolvedTheme): Shape =
-    if (modifiers.any { it is ProtocolModifier.Shape || it is ProtocolModifier.ShapeRole || it is ProtocolModifier.CornerEach }) {
+    if (setsOwnShape()) {
         modifiers.resolvedShape(theme)
     } else {
         style.shape
@@ -98,12 +107,34 @@ internal fun Modifier.containerDecoration(
     theme: ResolvedTheme,
     /** True for a menu, a dialog or a tooltip: a sheet over the page rather than in it. */
     overlay: Boolean = false,
+    role: ContainerRole? = null,
 ): Modifier {
-    val shape = node.containerShape(style, theme)
-    val raised = if (node.setsOwnElevation()) {
+    val shape = if (node.setsOwnShape()) {
+        node.containerShape(style, theme)
+    } else if (overlay) {
+        when (role) {
+            ContainerRole.Dialog -> theme.overlay?.dialogCornerRadius?.takeIf { it > 0 }?.let { RoundedCornerShape(it.dp) }
+            ContainerRole.Menu -> theme.overlay?.popupCornerRadius?.takeIf { it > 0 }?.let { RoundedCornerShape(it.dp) }
+            else -> null
+        } ?: style.shape
+    } else {
+        style.shape
+    }
+    val effectiveElevation = if (node.setsOwnElevation()) {
+        0.dp
+    } else if (overlay) {
+        when (role) {
+            ContainerRole.Dialog -> theme.overlay?.dialogElevation?.takeIf { it > 0 }?.let { it.dp }
+            ContainerRole.Menu -> theme.overlay?.popupElevation?.takeIf { it > 0 }?.let { it.dp }
+            else -> null
+        } ?: style.elevation
+    } else {
+        style.elevation
+    }
+    val raised = if (node.setsOwnElevation() || effectiveElevation.value <= 0f) {
         this
     } else {
-        theme.rules.elevation(this, style.elevation, shape, theme)
+        theme.rules.elevation(this, effectiveElevation, shape, theme)
     }
     val material = style.material
     // A Host that named its own Background has already had it applied, so painting the
@@ -139,6 +170,17 @@ internal fun Modifier.containerDecoration(
         .then(
             if (node.setsOwnPadding()) {
                 Modifier
+            } else if (overlay) {
+                val customPadding = when (role) {
+                    ContainerRole.Dialog -> theme.overlay?.dialogPadding?.takeIf { it > 0 }?.let { it.dp }
+                    ContainerRole.Menu -> theme.overlay?.popupPadding?.takeIf { it > 0 }?.let { it.dp }
+                    else -> null
+                }
+                if (customPadding != null) {
+                    Modifier.padding(customPadding)
+                } else {
+                    Modifier.padding(horizontal = style.horizontalPadding, vertical = style.verticalPadding)
+                }
             } else {
                 Modifier.padding(horizontal = style.horizontalPadding, vertical = style.verticalPadding)
             },
diff --git a/compose-rust/android-kotlin/shared/HostDropdown.kt b/compose-rust/android-kotlin/shared/HostDropdown.kt
index 52b3f526..dc1fcdd9 100644
--- a/compose-rust/android-kotlin/shared/HostDropdown.kt
+++ b/compose-rust/android-kotlin/shared/HostDropdown.kt
@@ -196,6 +196,29 @@ private fun OpeningList(
             )
         }
         if (open) {
+            val popupShape = if (node.setsOwnShape()) {
+                node.containerShape(style, theme)
+            } else {
+                theme.overlay?.popupCornerRadius?.takeIf { it > 0 }?.let { androidx.compose.foundation.shape.RoundedCornerShape(it.dp) } ?: style.shape
+            }
+            val popupElevation = if (node.setsOwnElevation()) {
+                0.dp
+            } else {
+                theme.overlay?.popupElevation?.takeIf { it > 0 }?.let { it.dp } ?: style.elevation
+            }
+            val popupRaised = if (node.setsOwnElevation() || popupElevation.value <= 0f) {
+                Modifier
+            } else {
+                theme.rules.elevation(Modifier, popupElevation, popupShape, theme)
+            }
+            val explicitPopupPad = theme.overlay?.popupPadding
+            val popupPadding = if (node.setsOwnPadding()) {
+                Modifier
+            } else if (explicitPopupPad != null && explicitPopupPad > 0) {
+                Modifier.padding(explicitPopupPad.dp)
+            } else {
+                Modifier.padding(vertical = theme.space(SpaceRole.Xs))
+            }
             Popup(
                 popupPositionProvider = BelowAnchorPositionProvider,
                 onDismissRequest = { open = false },
@@ -205,10 +228,11 @@ private fun OpeningList(
                     Column(
                         Modifier
                             .testTag(dropdownPopupTestTag(node.id))
-                            .clip(style.shape)
-                            .background(style.container, style.shape)
-                            .border(style.borderWidth, style.borderColor, style.shape)
-                            .padding(vertical = theme.space(SpaceRole.Xs)),
+                            .then(popupRaised)
+                            .clip(popupShape)
+                            .background(style.container, popupShape)
+                            .border(style.borderWidth, style.borderColor, popupShape)
+                            .then(popupPadding),
                     ) {
                         options.forEachIndexed { index, childId ->
                             key(childId) {
diff --git a/compose-rust/android-kotlin/shared/HostOverlay.kt b/compose-rust/android-kotlin/shared/HostOverlay.kt
index 43483ea3..834c0246 100644
--- a/compose-rust/android-kotlin/shared/HostOverlay.kt
+++ b/compose-rust/android-kotlin/shared/HostOverlay.kt
@@ -50,6 +50,7 @@ import dev.darkpyonix.composerust.ui.node.Node
 import dev.darkpyonix.composerust.ui.node.NodeTable
 import dev.darkpyonix.composerust.ui.node.RenderNode
 import dev.darkpyonix.composerust.ui.node.nodeTestTag
+import dev.darkpyonix.composerust.ui.paintProp
 import dev.darkpyonix.composerust.ui.textStyle
 import dev.darkpyonix.composerust.design.LocalGlassBackdrop
 import androidx.compose.runtime.CompositionLocalProvider
@@ -194,6 +195,9 @@ internal fun HostDialog(
     val open = rememberOverlayOpen(node)
     if (!open.value) return
     val style = theme.rules.container(ContainerRole.Dialog, theme)
+    val scrim = node.paintProp(PropertyKind.Scrim)?.let { theme.color(it) }
+        ?: theme.overlay?.scrim?.takeIf { it != 0L }?.let { androidx.compose.ui.graphics.Color(it.toULong()) }
+        ?: style.scrim
     val surfaceRequester = remember(node.id) { FocusRequester() }
     val onEscape = { dismiss(node, dispatcher) }
     Popup(
@@ -206,14 +210,13 @@ internal fun HostDialog(
     ) {
         ModalOverlayFocus(node.id, surfaceRequester, onEscape)
         Box(
-            Modifier.fillMaxSize().background(style.scrim),
+            Modifier.fillMaxSize().background(scrim).testTag(dialogTestTag(node.id)),
             contentAlignment = Alignment.Center,
         ) {
             Column(
                 modifier
-                    .testTag(dialogTestTag(node.id))
                     .modalOverlaySurface(node.id, surfaceRequester)
-                    .containerDecoration(node, style, theme, overlay = true),
+                    .containerDecoration(node, style, theme, overlay = true, role = ContainerRole.Dialog),
             ) {
                 node.children.forEach { childId ->
                     key(childId) { RenderNode(childId, table, dispatcher) }
@@ -265,7 +268,7 @@ internal fun HostMenu(
                 Column(
                     Modifier
                         .testTag(menuPopupTestTag(node.id))
-                        .containerDecoration(node, style, theme, overlay = true),
+                        .containerDecoration(node, style, theme, overlay = true, role = ContainerRole.Menu),
                 ) {
                     entries.forEach { childId ->
                         key(childId) { RenderNode(childId, table, dispatcher) }
diff --git a/compose-rust/android-kotlin/shared/HostSheet.kt b/compose-rust/android-kotlin/shared/HostSheet.kt
index f5a12bb9..74cd7dce 100644
--- a/compose-rust/android-kotlin/shared/HostSheet.kt
+++ b/compose-rust/android-kotlin/shared/HostSheet.kt
@@ -34,6 +34,7 @@ import androidx.compose.ui.window.Popup
 import androidx.compose.ui.window.PopupProperties
 import dev.darkpyonix.composerust.design.ResolvedTheme
 import dev.darkpyonix.composerust.design.SheetEdge
+import dev.darkpyonix.composerust.protocol.PropertyKind
 import dev.darkpyonix.composerust.protocol.ShapeRole
 import dev.darkpyonix.composerust.runtime.EventDispatcher
 import dev.darkpyonix.composerust.runtime.LocalWindowSizeClass
@@ -41,6 +42,7 @@ import dev.darkpyonix.composerust.ui.node.Node
 import dev.darkpyonix.composerust.ui.node.NodeTable
 import dev.darkpyonix.composerust.ui.node.RenderNode
 import dev.darkpyonix.composerust.ui.node.nodeTestTag
+import dev.darkpyonix.composerust.ui.paintProp
 import kotlin.math.roundToInt
 
 /** How far a sheet has to be pulled towards its edge before letting go closes it. */
@@ -84,6 +86,65 @@ internal fun HostSheet(
     val surfaceRequester = remember(node.id) { FocusRequester() }
     val onEscape = { dismiss(node, dispatcher) }
 
+    val scrim = node.paintProp(PropertyKind.Scrim)?.let { theme.color(it) }
+        ?: theme.overlay?.scrim?.takeIf { it != 0L }?.let { androidx.compose.ui.graphics.Color(it.toULong()) }
+        ?: style.scrim
+
+    val shape = if (node.setsOwnShape()) {
+        node.containerShape(
+            dev.darkpyonix.composerust.design.ContainerStyle(
+                container = style.container,
+                content = androidx.compose.ui.graphics.Color.Unspecified,
+                shape = style.shape,
+                elevation = style.elevation,
+                borderWidth = style.borderWidth,
+                borderColor = style.borderColor,
+                horizontalPadding = 0.dp,
+                verticalPadding = 0.dp,
+                separator = null,
+                scrim = style.scrim,
+                typeRole = dev.darkpyonix.composerust.protocol.TypeRole.Body,
+            ),
+            theme,
+        )
+    } else {
+        theme.overlay?.sheetCornerRadius?.takeIf { it > 0 }?.let {
+            if (bottom) {
+                androidx.compose.foundation.shape.RoundedCornerShape(it.dp, it.dp, 0.dp, 0.dp)
+            } else {
+                androidx.compose.foundation.shape.RoundedCornerShape(it.dp, 0.dp, 0.dp, it.dp)
+            }
+        } ?: style.shape
+    }
+
+    val sheetElevation = if (node.setsOwnElevation()) {
+        0.dp
+    } else {
+        theme.overlay?.sheetElevation?.takeIf { it > 0 }?.let { it.dp } ?: style.elevation
+    }
+    val raised = if (node.setsOwnElevation() || sheetElevation.value <= 0f) {
+        Modifier
+    } else {
+        theme.rules.elevation(Modifier, sheetElevation, shape, theme)
+    }
+
+    val containerBg = if (node.setsOwnBackground()) Modifier else Modifier.background(style.container)
+
+    val outline = if (style.borderWidth.value > 0f) {
+        Modifier.border(style.borderWidth, style.borderColor, shape)
+    } else {
+        Modifier
+    }
+
+    val explicitSheetPad = theme.overlay?.sheetPadding
+    val sheetPadding = if (node.setsOwnPadding()) {
+        Modifier
+    } else if (explicitSheetPad != null && explicitSheetPad > 0) {
+        Modifier.padding(explicitSheetPad.dp)
+    } else {
+        Modifier.padding(style.padding)
+    }
+
     Popup(
         alignment = Alignment.Center,
         properties = PopupProperties(focusable = true, dismissOnBackPress = false),
@@ -94,7 +155,7 @@ internal fun HostSheet(
             Modifier
                 .testTag(sheetScrimTestTag(node.id))
                 .fillMaxSize()
-                .background(style.scrim)
+                .background(scrim)
                 // Pressing what the sheet covers is the other way of asking to close it,
                 // and it gets the same answer: one dismissal. A raw tap detector rather
                 // than `clickable`, because a scrim is not a button and saying it is one
@@ -107,11 +168,6 @@ internal fun HostSheet(
             } else {
                 Modifier.fillMaxHeight().fillMaxWidth(style.widthFraction)
             }
-            val outline = if (style.borderWidth.value > 0f) {
-                Modifier.border(style.borderWidth, style.borderColor, style.shape)
-            } else {
-                Modifier
-            }
             // Two boxes rather than one: the outer is the sheet itself, which is what is
             // dragged and what the design system sized, and the inner carries whatever
             // Modifiers the Host put on the node, which belong inside its surface.
@@ -127,8 +183,9 @@ internal fun HostSheet(
                             IntOffset(travel.roundToInt(), 0)
                         }
                     }
-                    .clip(style.shape)
-                    .background(style.container)
+                    .then(raised)
+                    .clip(shape)
+                    .then(containerBg)
                     .then(outline)
                     .draggable(
                         orientation = if (bottom) Orientation.Vertical else Orientation.Horizontal,
@@ -143,7 +200,7 @@ internal fun HostSheet(
                     // swallowed here rather than reaching the scrim behind.
                     .pointerInput(Unit) { detectTapGestures { } },
             ) {
-                Column(modifier.fillMaxWidth().padding(style.padding)) {
+                Column(modifier.fillMaxWidth().then(sheetPadding)) {
                     style.handle?.let { handle ->
                         Box(Modifier.fillMaxWidth(), contentAlignment = Alignment.Center) {
                             Box(
diff --git a/compose-rust/android-kotlin/shared/NodeTable.kt b/compose-rust/android-kotlin/shared/NodeTable.kt
index 1d4dab3f..e74429f5 100644
--- a/compose-rust/android-kotlin/shared/NodeTable.kt
+++ b/compose-rust/android-kotlin/shared/NodeTable.kt
@@ -8,6 +8,7 @@ import androidx.compose.runtime.setValue
 import dev.darkpyonix.composerust.foundation.ChromeInsets
 import dev.darkpyonix.composerust.foundation.MessageQueue
 import dev.darkpyonix.composerust.protocol.Mutation
+import dev.darkpyonix.composerust.protocol.OverlayTokens
 import dev.darkpyonix.composerust.protocol.PropertyKind
 import dev.darkpyonix.composerust.protocol.PropertyValue
 import dev.darkpyonix.composerust.protocol.Theme
@@ -187,6 +188,10 @@ class NodeTable {
     var theme: Theme? by mutableStateOf(null)
         private set
 
+    /** Null until the Host sends an overlay tokens record. */
+    var overlayTokens: OverlayTokens? by mutableStateOf(null)
+        private set
+
     /** Null until the Host sends its first `SetWindow` record. */
     var window: Window? by mutableStateOf(null)
         private set
@@ -320,6 +325,7 @@ class NodeTable {
                     fail(TableError.INVALID_PALETTE, "theme palette: $problem")
                 }
             }
+            is Mutation.SetOverlayTokens -> overlayTokens = mutation.overlay
             // Read before there is a window to apply it to. The platform layer asks for
             // this out of the first batch and builds the window from it, so by the time
             // the batch is applied the window already looks the way it says. Keeping it
@@ -974,6 +980,11 @@ class NodeTable {
                     widget == WidgetKind.Menu ||
                     // A sheet is an overlay on the same terms as a dialog.
                     widget == WidgetKind.Sheet
+
+                // A custom scrim colour for modal and popup overlays.
+                PropertyKind.Scrim -> widget == WidgetKind.Dialog ||
+                    widget == WidgetKind.Menu ||
+                    widget == WidgetKind.Sheet
                 // A Dropdown's selection is a position in its own children, the same thing
                 // a tab strip's selection is.
                 PropertyKind.SelectedIndex ->
diff --git a/compose-rust/android-kotlin/shared/Protocol.gen.kt b/compose-rust/android-kotlin/shared/Protocol.gen.kt
index a3a53b00..93583d78 100644
--- a/compose-rust/android-kotlin/shared/Protocol.gen.kt
+++ b/compose-rust/android-kotlin/shared/Protocol.gen.kt
@@ -8,7 +8,7 @@ import java.nio.charset.StandardCharsets
 
 enum class WidgetKind { Column, Row, Box, Text, TextField, Button, Spacer, LazyColumn, ScrollColumn, Image, Icon, Checkbox, RadioButton, Switch, Slider, ProgressIndicator, Divider, Card, Surface, Dialog, Menu, Tabs, TopAppBar, LazyRow, Tooltip, Canvas, DatePicker, TimePicker, Dropdown, Navigation, NavigationItem, Sheet, Scaffold, ScaffoldSlot, LazyGrid, FileDropTarget, ScrollRow, Chip, FloatingAction, Badge, SelectionContainer, SplitPane, CodeEditor, AbsoluteBox, Window, LinearProgressIndicator }
 
-enum class PropertyKind { Text, Placeholder, Enabled, Multiline, OnClick, OnValueChange, OnSubmit, OnFocusLost, OnKeyDown, ItemCount, ItemKey, OnRangeRequested, TypeRole, FontSize, FontWeight, LineHeight, LetterSpacing, Color, TextAlign, MaxLines, Overflow, Arrangement, Spacing, SpaceRole, Alignment, Variant, Asset, Checked, Steps, Determinate, Circular, Vertical, Open, OnDismiss, SelectedIndex, Commands, Value, Min, Max, Icon, Slot, Columns, MinColumnWidth, Spans, OnFilesEntered, OnFilesDropped, Section, Count, Collapsible, ButtonKind, Decorations, SyntaxSpans, TabWidth, OnEditRejected, OnHover, OnSave, OnDecorationClick, Folds, GutterDecorations, OnSelectionChange, Font, WordBreak, OverflowWrap, TabSize, AbsoluteSize, SoftWrap, SpanFonts, OnVisibleRangeChange, OnEditorFocusChange, OnFoldChange, OnCompositionChange, OnPointer, PointerKinds, Cursor, UserSelect, IslandZoom, WindowTitle, WindowSize, WindowMinSize, WindowResizable, WindowChrome, WindowTitleBar, WindowIcon, WindowPosition, WindowRestoreKey, WindowZoomLevel, DragFromPointer, OnWindowCloseRequested, OnWindowFocusChanged, OnWindowMoved, OnWindowDrag, OnWindowUnavailable, OnWindowClosed, OnWindowDragOver, TearOffKey, OnFocusGained, OnKeyUp, AccessibleName, TrackColor, ThumbColor, ActiveTrackColor, ContainerColor, CheckColor, ControlSize, ReportSize, Progress }
+enum class PropertyKind { Text, Placeholder, Enabled, Multiline, OnClick, OnValueChange, OnSubmit, OnFocusLost, OnKeyDown, ItemCount, ItemKey, OnRangeRequested, TypeRole, FontSize, FontWeight, LineHeight, LetterSpacing, Color, TextAlign, MaxLines, Overflow, Arrangement, Spacing, SpaceRole, Alignment, Variant, Asset, Checked, Steps, Determinate, Circular, Vertical, Open, OnDismiss, SelectedIndex, Commands, Value, Min, Max, Icon, Slot, Columns, MinColumnWidth, Spans, OnFilesEntered, OnFilesDropped, Section, Count, Collapsible, ButtonKind, Decorations, SyntaxSpans, TabWidth, OnEditRejected, OnHover, OnSave, OnDecorationClick, Folds, GutterDecorations, OnSelectionChange, Font, WordBreak, OverflowWrap, TabSize, AbsoluteSize, SoftWrap, SpanFonts, OnVisibleRangeChange, OnEditorFocusChange, OnFoldChange, OnCompositionChange, OnPointer, PointerKinds, Cursor, UserSelect, IslandZoom, WindowTitle, WindowSize, WindowMinSize, WindowResizable, WindowChrome, WindowTitleBar, WindowIcon, WindowPosition, WindowRestoreKey, WindowZoomLevel, DragFromPointer, OnWindowCloseRequested, OnWindowFocusChanged, OnWindowMoved, OnWindowDrag, OnWindowUnavailable, OnWindowClosed, OnWindowDragOver, TearOffKey, OnFocusGained, OnKeyUp, AccessibleName, TrackColor, ThumbColor, ActiveTrackColor, ContainerColor, CheckColor, ControlSize, ReportSize, Scrim, Progress }
 
 enum class Key { Enter, Other }
 
@@ -195,6 +195,20 @@ data class Theme(
 /** One role's colour in one scheme, as the application gave it. */
 data class PaletteEntry(val role: ColorRole, val dark: Boolean, val argb: Int)
 
+/** Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+data class OverlayTokens(
+    val sheetCornerRadius: Int = 0,
+    val dialogCornerRadius: Int = 0,
+    val popupCornerRadius: Int = 0,
+    val sheetElevation: Int = 0,
+    val popupElevation: Int = 0,
+    val dialogElevation: Int = 0,
+    val sheetPadding: Int = 0,
+    val dialogPadding: Int = 0,
+    val popupPadding: Int = 0,
+    val scrim: Long = 0L,
+)
+
 /**
  * What the application asked of its own window.
  *
@@ -1279,6 +1293,9 @@ sealed interface Mutation {
         val name: String,
         val filters: String,
     ) : Mutation
+
+    /** Sets application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+    data class SetOverlayTokens(val overlay: OverlayTokens) : Mutation
 }
 
 sealed interface HostEvent {
@@ -1334,7 +1351,7 @@ class ProtocolException(message: String, val offset: Int) :
     IllegalArgumentException("$message at byte offset $offset")
 
 object Protocol {
-    const val SCHEMA_HASH: Long = 6134367089496884194L
+    const val SCHEMA_HASH: Long = 2307080319921313708L
     const val PROTOCOL_VERSION: Int = 1
 
     private const val TAG_ENVELOPE = 0
@@ -1364,6 +1381,7 @@ object Protocol {
     private const val TAG_FOCUS_WINDOW = 24
     private const val TAG_REQUEST_FOCUS = 25
     private const val TAG_SHOW_FILE_DIALOG = 26
+    private const val TAG_SET_OVERLAY_TOKENS = 27
     private const val ENVELOPE_LENGTH = 12
     private const val ANIMATION_HEADER_LENGTH = 52
     private const val KEYFRAME_HEAD_LENGTH = 24
@@ -1530,6 +1548,23 @@ object Protocol {
                             readString(batch, base, available, offset + 40),
                         )
                     }
+                    TAG_SET_OVERLAY_TOKENS -> {
+                        requireRecordLength(length, 28, offset)
+                        Mutation.SetOverlayTokens(
+                            OverlayTokens(
+                                sheetCornerRadius = readU16(batch, base, available, offset + 4),
+                                dialogCornerRadius = readU16(batch, base, available, offset + 6),
+                                popupCornerRadius = readU16(batch, base, available, offset + 8),
+                                sheetElevation = readU16(batch, base, available, offset + 10),
+                                popupElevation = readU16(batch, base, available, offset + 12),
+                                dialogElevation = readU16(batch, base, available, offset + 14),
+                                sheetPadding = readU16(batch, base, available, offset + 16),
+                                dialogPadding = readU16(batch, base, available, offset + 18),
+                                popupPadding = readU16(batch, base, available, offset + 20),
+                                scrim = readU32(batch, base, available, offset + 24),
+                            ),
+                        )
+                    }
                     TAG_SET_WINDOW -> {
                         requireRecordLength(length, 32, offset)
                         // Bit 0 is `resizable`, bit 1 asks for every whole-dp size change; the
@@ -2283,6 +2318,7 @@ object Protocol {
         125 -> PropertyKind.CheckColor
         126 -> PropertyKind.ControlSize
         127 -> PropertyKind.ReportSize
+        128 -> PropertyKind.Scrim
         27 -> PropertyKind.Progress
         else -> throw ProtocolException("unknown property tag $tag", offset)
     }
diff --git a/compose-rust/schema-hash.txt b/compose-rust/schema-hash.txt
index b543eb68..03b6c45d 100644
--- a/compose-rust/schema-hash.txt
+++ b/compose-rust/schema-hash.txt
@@ -1 +1 @@
-0x5521a65ddb4f03e2
+0x20046553d93457ac
diff --git a/compose-rust/src/codegen.rs b/compose-rust/src/codegen.rs
index 918a18c0..da7e6dc6 100644
--- a/compose-rust/src/codegen.rs
+++ b/compose-rust/src/codegen.rs
@@ -148,6 +148,20 @@ data class Theme(
 /** One role's colour in one scheme, as the application gave it. */
 data class PaletteEntry(val role: ColorRole, val dark: Boolean, val argb: Int)
 
+/** Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+data class OverlayTokens(
+    val sheetCornerRadius: Int = 0,
+    val dialogCornerRadius: Int = 0,
+    val popupCornerRadius: Int = 0,
+    val sheetElevation: Int = 0,
+    val popupElevation: Int = 0,
+    val dialogElevation: Int = 0,
+    val sheetPadding: Int = 0,
+    val dialogPadding: Int = 0,
+    val popupPadding: Int = 0,
+    val scrim: Long = 0L,
+)
+
 /**
  * What the application asked of its own window.
  *
@@ -488,6 +502,9 @@ sealed interface Mutation {
         val name: String,
         val filters: String,
     ) : Mutation
+
+    /** Sets application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+    data class SetOverlayTokens(val overlay: OverlayTokens) : Mutation
 }
 
 "#,
@@ -626,6 +643,7 @@ object Protocol {
     private const val TAG_FOCUS_WINDOW = 24
     private const val TAG_REQUEST_FOCUS = 25
     private const val TAG_SHOW_FILE_DIALOG = 26
+    private const val TAG_SET_OVERLAY_TOKENS = 27
     private const val ENVELOPE_LENGTH = 12
     private const val ANIMATION_HEADER_LENGTH = 52
     private const val KEYFRAME_HEAD_LENGTH = 24
@@ -792,6 +810,23 @@ object Protocol {
                             readString(batch, base, available, offset + 40),
                         )
                     }
+                    TAG_SET_OVERLAY_TOKENS -> {
+                        requireRecordLength(length, 28, offset)
+                        Mutation.SetOverlayTokens(
+                            OverlayTokens(
+                                sheetCornerRadius = readU16(batch, base, available, offset + 4),
+                                dialogCornerRadius = readU16(batch, base, available, offset + 6),
+                                popupCornerRadius = readU16(batch, base, available, offset + 8),
+                                sheetElevation = readU16(batch, base, available, offset + 10),
+                                popupElevation = readU16(batch, base, available, offset + 12),
+                                dialogElevation = readU16(batch, base, available, offset + 14),
+                                sheetPadding = readU16(batch, base, available, offset + 16),
+                                dialogPadding = readU16(batch, base, available, offset + 18),
+                                popupPadding = readU16(batch, base, available, offset + 20),
+                                scrim = readU32(batch, base, available, offset + 24),
+                            ),
+                        )
+                    }
                     TAG_SET_WINDOW -> {
                         requireRecordLength(length, 32, offset)
                         // Bit 0 is `resizable`, bit 1 asks for every whole-dp size change; the
diff --git a/compose-rust/src/protocol.rs b/compose-rust/src/protocol.rs
index f87d19f1..a60644c5 100644
--- a/compose-rust/src/protocol.rs
+++ b/compose-rust/src/protocol.rs
@@ -42,6 +42,10 @@ const TAG_SHOW_FILE_DIALOG: u16 = 26;
 /// `ShowFileDialog` after its header: the request, the window, the kind and the default
 /// filter, then four string references. 44 bytes, so the record is 48.
 const SHOW_FILE_DIALOG_PAYLOAD_LEN: usize = 4 + 4 + 2 + 2 + 4 * 8;
+// 27 sets application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet.
+const TAG_SET_OVERLAY_TOKENS: u16 = 27;
+/// 9 u16 fields (18 bytes) + 1 u16 reserved (2 bytes) + 1 u32 scrim (4 bytes) = 24 bytes.
+const OVERLAY_TOKENS_PAYLOAD_LEN: usize = 24;
 /// The head of a `StartAnimation` record, before its keyframes.
 const ANIMATION_HEADER_LEN: usize = 52;
 /// A keyframe before its value: offset, timing, step position, flags and four numbers.
@@ -268,6 +272,8 @@ pub enum Mutation<'a> {
         name: &'a str,
         filters: &'a str,
     },
+    /// Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet.
+    SetOverlayTokens(crate::schema::OverlayTokens),
 }
 
 #[derive(Clone, Debug, Eq, PartialEq)]
@@ -1507,6 +1513,20 @@ impl BatchEncoder {
                 self.put_string_ref(name)?;
                 self.put_string_ref(filters)?;
             }
+            Mutation::SetOverlayTokens(overlay) => {
+                self.begin_record(TAG_SET_OVERLAY_TOKENS, OVERLAY_TOKENS_PAYLOAD_LEN as u16);
+                self.put_u16(overlay.sheet_corner_radius);
+                self.put_u16(overlay.dialog_corner_radius);
+                self.put_u16(overlay.popup_corner_radius);
+                self.put_u16(overlay.sheet_elevation);
+                self.put_u16(overlay.popup_elevation);
+                self.put_u16(overlay.dialog_elevation);
+                self.put_u16(overlay.sheet_padding);
+                self.put_u16(overlay.dialog_padding);
+                self.put_u16(overlay.popup_padding);
+                self.put_u16(0);
+                self.put_u32(overlay.scrim);
+            }
             Mutation::SetWindow(window) => {
                 self.begin_record(TAG_SET_WINDOW, 28);
                 self.put_u16(window.chrome as u16);
@@ -1827,6 +1847,7 @@ fn decode_theme(bytes: &[u8], payload: usize, records_len: usize) -> Result<Them
         adaptive: adaptive == 1,
         fonts,
         palette,
+        overlay: None,
     })
 }
 
@@ -1966,6 +1987,31 @@ pub fn decode_batch(bytes: &[u8]) -> Result<Vec<Mutation<'_>>, ProtocolError> {
                 }
             }
             TAG_SHOW_FILE_DIALOG => return Err(ProtocolError::InvalidRecordLength),
+            TAG_SET_OVERLAY_TOKENS if len == 4 + OVERLAY_TOKENS_PAYLOAD_LEN => {
+                let sheet_corner_radius = read_u16(bytes, payload)?;
+                let dialog_corner_radius = read_u16(bytes, payload + 2)?;
+                let popup_corner_radius = read_u16(bytes, payload + 4)?;
+                let sheet_elevation = read_u16(bytes, payload + 6)?;
+                let popup_elevation = read_u16(bytes, payload + 8)?;
+                let dialog_elevation = read_u16(bytes, payload + 10)?;
+                let sheet_padding = read_u16(bytes, payload + 12)?;
+                let dialog_padding = read_u16(bytes, payload + 14)?;
+                let popup_padding = read_u16(bytes, payload + 16)?;
+                let scrim = read_u32(bytes, payload + 20)?;
+                Mutation::SetOverlayTokens(crate::schema::OverlayTokens {
+                    sheet_corner_radius,
+                    dialog_corner_radius,
+                    popup_corner_radius,
+                    sheet_elevation,
+                    popup_elevation,
+                    dialog_elevation,
+                    sheet_padding,
+                    dialog_padding,
+                    popup_padding,
+                    scrim,
+                })
+            }
+            TAG_SET_OVERLAY_TOKENS => return Err(ProtocolError::InvalidRecordLength),
             TAG_SET_WINDOW if len == 32 => {
                 let chrome = read_u16(bytes, payload)?;
                 let title_bar = read_u16(bytes, payload + 2)?;
@@ -2136,7 +2182,7 @@ pub fn decode_batch(bytes: &[u8]) -> Result<Vec<Mutation<'_>>, ProtocolError> {
             | TAG_START_ANIMATION
             | TAG_CONTROL_ANIMATION
             | TAG_SET_SELECTION..=TAG_SET_FOLDED
-            | TAG_SET_WINDOW_THEME..=TAG_REQUEST_FOCUS => {
+            | TAG_SET_WINDOW_THEME..=TAG_SET_OVERLAY_TOKENS => {
                 return Err(ProtocolError::InvalidRecordLength);
             }
             other => return Err(ProtocolError::InvalidTag(other)),
diff --git a/compose-rust/src/runtime/seam.rs b/compose-rust/src/runtime/seam.rs
index ea62e8b6..cf9e7612 100644
--- a/compose-rust/src/runtime/seam.rs
+++ b/compose-rust/src/runtime/seam.rs
@@ -86,6 +86,14 @@ impl Batch {
     /// The root theme record. Written once per rebuild, never per frame.
     pub fn set_theme(&mut self, theme: Theme) {
         self.write(Mutation::SetTheme(theme));
+        if let Some(overlay) = theme.overlay {
+            self.write(Mutation::SetOverlayTokens(*overlay));
+        }
+    }
+
+    /// Sets application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet.
+    pub fn set_overlay_tokens(&mut self, tokens: crate::schema::OverlayTokens) {
+        self.write(Mutation::SetOverlayTokens(tokens));
     }
 
     /// The root window record. Written once per rebuild, never per frame.
diff --git a/compose-rust/src/schema.rs b/compose-rust/src/schema.rs
index bfbe3f08..dd90e7ed 100644
--- a/compose-rust/src/schema.rs
+++ b/compose-rust/src/schema.rs
@@ -206,13 +206,13 @@ pub struct EventSchema {
 pub const SCHEMA_DESCRIPTOR: &str = concat!(
     "compose-rust/v1;",
     "widgets=Column,Row,Box,Text,TextField,Button,Spacer,LazyColumn,ScrollColumn,Image,Icon,Checkbox,RadioButton,Switch,Slider,ProgressIndicator,Divider,Card,Surface,Dialog,Menu,Tabs,TopAppBar,LazyRow,Tooltip,Canvas,DatePicker,TimePicker,Dropdown,Navigation,NavigationItem,Sheet,Scaffold,ScaffoldSlot,LazyGrid,FileDropTarget,ScrollRow,Chip,FloatingAction,Badge,SelectionContainer,SplitPane,CodeEditor,AbsoluteBox,Window;",
-    "properties=text,placeholder,enabled,multiline,on_click,on_value_change,on_submit,on_focus_lost,on_key_down,item_count,item_key,on_range_requested,type_role,font_size,font_weight,line_height,letter_spacing,color,text_align,max_lines,overflow,arrangement,spacing,space_role,alignment,variant,asset,checked,steps,determinate,circular,vertical,open,on_dismiss,selected_index,commands,value,min,max,icon,slot,columns,min_column_width,spans,on_files_entered,on_files_dropped,section,count,collapsible,button_kind,decorations,syntax_spans,tab_width,on_edit_rejected,on_hover,on_save,on_decoration_click,folds,gutter_decorations,on_selection_change,font,word_break,overflow_wrap,tab_size,absolute_size,soft_wrap,span_fonts,on_visible_range_change,on_editor_focus_change,on_fold_change,on_composition_change,on_pointer,pointer_kinds,cursor,user_select,island_zoom,window_title,window_size,window_min_size,window_resizable,window_chrome,window_title_bar,window_icon,window_position,window_restore_key,window_zoom_level,drag_from_pointer,on_window_close_requested,on_window_focus_changed,on_window_moved,on_window_drag,on_window_unavailable,on_window_closed,on_window_drag_over,tear_off_key,on_focus_gained,on_key_up,accessible_name,track_color,thumb_color,active_track_color,container_color,check_color,control_size,report_size;",
+    "properties=text,placeholder,enabled,multiline,on_click,on_value_change,on_submit,on_focus_lost,on_key_down,item_count,item_key,on_range_requested,type_role,font_size,font_weight,line_height,letter_spacing,color,text_align,max_lines,overflow,arrangement,spacing,space_role,alignment,variant,asset,checked,steps,determinate,circular,vertical,open,on_dismiss,selected_index,commands,value,min,max,icon,slot,columns,min_column_width,spans,on_files_entered,on_files_dropped,section,count,collapsible,button_kind,decorations,syntax_spans,tab_width,on_edit_rejected,on_hover,on_save,on_decoration_click,folds,gutter_decorations,on_selection_change,font,word_break,overflow_wrap,tab_size,absolute_size,soft_wrap,span_fonts,on_visible_range_change,on_editor_focus_change,on_fold_change,on_composition_change,on_pointer,pointer_kinds,cursor,user_select,island_zoom,window_title,window_size,window_min_size,window_resizable,window_chrome,window_title_bar,window_icon,window_position,window_restore_key,window_zoom_level,drag_from_pointer,on_window_close_requested,on_window_focus_changed,on_window_moved,on_window_drag,on_window_unavailable,on_window_closed,on_window_drag_over,tear_off_key,on_focus_gained,on_key_up,accessible_name,track_color,thumb_color,active_track_color,container_color,check_color,control_size,report_size,scrim;",
     "modifiers=Empty,Padding,FillMaxWidth,FillMaxHeight,Width,Height,Size,Background,Clickable,PaddingRole,PaddingEach,Weight,Shape,ShapeRole,Border,Elevation,ObserveSize,Motion,Material,Offset,RequiredSize,BorderEach,CornerEach,Shadow,Clip,Alpha,Transform,Transform3D,MatchParentSize,WindowDrag,PressFeedback,Semantics;",
     "keys=Enter,Other;",
     "events=Clicked,TextChanged,TextSubmitted,FocusLost,ProtocolError,KeyDown,RangeRequested,ValueChanged,WindowSizeChanged,DesignSystemResolved,FilesEntered,FilesDropped,NotificationActivated,NotificationPermissionChanged,CodeChanged,CodeEditRejected,CodeHovered,CodeSaveRequested,DecorationActivated,AnimationEvent,ReducedMotionChanged,Pointer,ZoomChanged,SelectionChanged,VisibleRangeChanged,EditorFocusChanged,FoldChanged,CompositionChanged,WindowCloseRequested,WindowFocusChanged,WindowMoved,WindowDrag,WindowDragOver,WindowUnavailable,WindowClosed,WindowRequested,ApplicationReopen,FocusGained,KeyUp,FileDialogResult;",
     "windowsizeclasses=Compact,Medium,Expanded;",
     "windowheightclasses=Compact,Medium,Expanded;",
-    "commands=Create,SetProp,SetModifier,Insert,Move,Remove,SetText,AppendText,SetTheme,SetWindow,RegisterAsset,ReleaseAsset,ShowMessage,PostNotification,WithdrawNotification,RequestNotificationPermission,EditCode,StartAnimation,ControlAnimation,SetSelection,RevealRange,SetFolded,SetWindowTheme,FocusWindow,RequestFocus,ShowFileDialog"
+    "commands=Create,SetProp,SetModifier,Insert,Move,Remove,SetText,AppendText,SetTheme,SetWindow,RegisterAsset,ReleaseAsset,ShowMessage,PostNotification,WithdrawNotification,RequestNotificationPermission,EditCode,StartAnimation,ControlAnimation,SetSelection,RevealRange,SetFolded,SetWindowTheme,FocusWindow,RequestFocus,ShowFileDialog,SetOverlayTokens"
 );
 
 const fn hash_bytes(mut hash: u64, bytes: &[u8]) -> u64 {
@@ -1797,11 +1797,100 @@ pub struct Theme {
     /// platform. A palette built while the application runs is handed to `use_theme`,
     /// which owns it, so application code never has to give it that lifetime itself.
     pub palette: Option<&'static crate::palette::Palette>,
+    /// Application-level overlay theme tokens for Dialog, Menu, Dropdown, and Sheet.
+    pub overlay: Option<&'static OverlayTokens>,
 }
 
 /// How many type roles there are, which is how many font slots a theme carries.
 pub const TYPE_ROLE_COUNT: usize = 9;
 
+/// Application-level overlay theme tokens for Dialog, Menu, Dropdown, and Sheet.
+#[derive(Clone, Copy, Debug, Default, Eq, PartialEq)]
+pub struct OverlayTokens {
+    pub sheet_corner_radius: u16,
+    pub dialog_corner_radius: u16,
+    pub popup_corner_radius: u16,
+    pub sheet_elevation: u16,
+    pub popup_elevation: u16,
+    pub dialog_elevation: u16,
+    pub sheet_padding: u16,
+    pub dialog_padding: u16,
+    pub popup_padding: u16,
+    pub scrim: u32,
+}
+
+impl OverlayTokens {
+    pub const fn new() -> Self {
+        Self {
+            sheet_corner_radius: 0,
+            dialog_corner_radius: 0,
+            popup_corner_radius: 0,
+            sheet_elevation: 0,
+            popup_elevation: 0,
+            dialog_elevation: 0,
+            sheet_padding: 0,
+            dialog_padding: 0,
+            popup_padding: 0,
+            scrim: 0,
+        }
+    }
+
+    pub const fn sheet_corner_radius(mut self, radius: u16) -> Self {
+        self.sheet_corner_radius = radius;
+        self
+    }
+
+    pub const fn dialog_corner_radius(mut self, radius: u16) -> Self {
+        self.dialog_corner_radius = radius;
+        self
+    }
+
+    pub const fn popup_corner_radius(mut self, radius: u16) -> Self {
+        self.popup_corner_radius = radius;
+        self
+    }
+
+    pub const fn sheet_elevation(mut self, elevation: u16) -> Self {
+        self.sheet_elevation = elevation;
+        self
+    }
+
+    pub const fn popup_elevation(mut self, elevation: u16) -> Self {
+        self.popup_elevation = elevation;
+        self
+    }
+
+    pub const fn dialog_elevation(mut self, elevation: u16) -> Self {
+        self.dialog_elevation = elevation;
+        self
+    }
+
+    pub const fn sheet_padding(mut self, padding: u16) -> Self {
+        self.sheet_padding = padding;
+        self
+    }
+
+    pub const fn dialog_padding(mut self, padding: u16) -> Self {
+        self.dialog_padding = padding;
+        self
+    }
+
+    pub const fn popup_padding(mut self, padding: u16) -> Self {
+        self.popup_padding = padding;
+        self
+    }
+
+    pub const fn scrim(mut self, color: Color) -> Self {
+        self.scrim = color.to_argb();
+        self
+    }
+
+    pub const fn scrim_argb(mut self, argb: u32) -> Self {
+        self.scrim = argb;
+        self
+    }
+}
+
 impl Theme {
     /// The same design system on every platform.
     pub const fn unified(design_system: DesignSystem) -> Self {
@@ -1812,6 +1901,7 @@ impl Theme {
             adaptive: false,
             fonts: [0; TYPE_ROLE_COUNT],
             palette: None,
+            overlay: None,
         }
     }
 
@@ -1824,6 +1914,7 @@ impl Theme {
             adaptive: true,
             fonts: [0; TYPE_ROLE_COUNT],
             palette: None,
+            overlay: None,
         }
     }
 
@@ -1879,6 +1970,12 @@ impl Theme {
         self
     }
 
+    /// Overrides overlay geometry, spacing, elevation and scrim for Dialog, Menu, Dropdown and Sheet.
+    pub const fn with_overlay_tokens(mut self, overlay: &'static OverlayTokens) -> Self {
+        self.overlay = Some(overlay);
+        self
+    }
+
     /// The font this role resolves to, or `None` for the system font.
     pub const fn font(&self, role: TypeRole) -> Option<u32> {
         if let TypeRole::None = role {
@@ -3159,6 +3256,8 @@ crate::extensions::define_property_schema_with_extensions!(define_wire_enum; PRO
     // whole dp, changes, and not only when the size class does. A `Window` node's boolean;
     // absent is false.
     ReportSize = 127,
+    // Scrim colour for modal overlays (Dialog, Sheet, Menu), as a `Paint` bit value.
+    Scrim = 128,
 });
 
 #[derive(Clone, Debug, PartialEq)]
diff --git a/compose-rust/src/ui/widgets.rs b/compose-rust/src/ui/widgets.rs
index 8b4643c1..b7f5a52c 100644
--- a/compose-rust/src/ui/widgets.rs
+++ b/compose-rust/src/ui/widgets.rs
@@ -2211,6 +2211,7 @@ pub struct ModalScope {
     modifier: Modifier,
     widget: WidgetKind,
     open: bool,
+    scrim: Option<Paint>,
     on_dismiss: Option<Box<dyn FnMut()>>,
     done: bool,
 }
@@ -2222,6 +2223,7 @@ pub fn Dialog(open: bool) -> ModalScope {
         modifier: Modifier::new(),
         widget: WidgetKind::Dialog,
         open,
+        scrim: None,
         on_dismiss: None,
         done: false,
     }
@@ -2234,6 +2236,7 @@ pub fn Sheet(open: bool) -> ModalScope {
         modifier: Modifier::new(),
         widget: WidgetKind::Sheet,
         open,
+        scrim: None,
         on_dismiss: None,
         done: false,
     }
@@ -2245,6 +2248,12 @@ impl ModalScope {
         self
     }
 
+    /// Sets the modal scrim colour, overriding the design system default.
+    pub fn scrim(mut self, paint: impl Into<Paint>) -> Self {
+        self.scrim = Some(paint.into());
+        self
+    }
+
     fn compose(&mut self, content: impl FnOnce()) {
         let modifier = self.modifier;
         let (widget, open) = (self.widget, self.open);
@@ -2253,12 +2262,16 @@ impl ModalScope {
         } else {
             widget_key("compose_rust::ui::Sheet")
         };
+        let scrim = self.scrim;
         emit(
             key,
             widget,
             |attrs| {
                 push_modifiers(&modifier, Mods::Standard, attrs);
                 attrs.push(("open", OwnedAttr::Bool(open)));
+                if let Some(scrim) = scrim {
+                    attrs.push(("scrim", super::paint(Some(scrim))));
+                }
             },
             vec![("ondismiss", unit(self.on_dismiss.take()))],
             content,
@@ -2272,6 +2285,7 @@ container!(ModalScope);
 pub struct MenuScope<'a> {
     modifier: Modifier,
     expanded: bool,
+    scrim: Option<Paint>,
     on_dismiss: Option<Box<dyn FnMut()>>,
     anchor: Option<Box<dyn FnOnce() + 'a>>,
     done: bool,
@@ -2283,6 +2297,7 @@ pub fn Menu<'a>(expanded: bool) -> MenuScope<'a> {
     MenuScope {
         modifier: Modifier::new(),
         expanded,
+        scrim: None,
         on_dismiss: None,
         anchor: None,
         done: false,
@@ -2295,6 +2310,12 @@ impl<'a> MenuScope<'a> {
         self
     }
 
+    /// Sets the popup scrim colour, overriding the design system default.
+    pub fn scrim(mut self, paint: impl Into<Paint>) -> Self {
+        self.scrim = Some(paint.into());
+        self
+    }
+
     /// The widget the menu hangs off. It is the first child on the wire.
     pub fn anchor(mut self, anchor: impl FnOnce() + 'a) -> Self {
         self.anchor = Some(std::boxed::Box::new(anchor));
@@ -2305,12 +2326,16 @@ impl<'a> MenuScope<'a> {
         let modifier = self.modifier;
         let expanded = self.expanded;
         let anchor = self.anchor.take();
+        let scrim = self.scrim;
         emit(
             widget_key("compose_rust::ui::Menu"),
             WidgetKind::Menu,
             |attrs| {
                 push_modifiers(&modifier, Mods::Standard, attrs);
                 attrs.push(("open", OwnedAttr::Bool(expanded)));
+                if let Some(scrim) = scrim {
+                    attrs.push(("scrim", super::paint(Some(scrim))));
+                }
             },
             vec![("ondismiss", unit(self.on_dismiss.take()))],
             || {
diff --git a/compose-rust/tests/fr74_overlay_tokens.rs b/compose-rust/tests/fr74_overlay_tokens.rs
new file mode 100644
index 00000000..2b5e5d51
--- /dev/null
+++ b/compose-rust/tests/fr74_overlay_tokens.rs
@@ -0,0 +1,70 @@
+//! Overlay theme tokens and node geometry overrides.
+
+use compose_rust::protocol::{BatchEncoder, Mutation, decode_batch};
+use compose_rust::schema::{DesignSystem, OverlayTokens, PropertyKind, Theme};
+
+#[test]
+fn fr74_overlay_tokens_command_round_trips() {
+    let tokens = OverlayTokens::new()
+        .sheet_corner_radius(24)
+        .dialog_corner_radius(16)
+        .popup_corner_radius(12)
+        .sheet_elevation(8)
+        .dialog_elevation(6)
+        .popup_elevation(4)
+        .sheet_padding(20)
+        .dialog_padding(16)
+        .popup_padding(8)
+        .scrim_argb(0x80000000);
+
+    let mut encoder = BatchEncoder::default();
+    encoder.encode(&Mutation::SetOverlayTokens(tokens)).unwrap();
+    let bytes = encoder.finish().unwrap();
+
+    let decoded = decode_batch(bytes).unwrap();
+    assert_eq!(decoded.len(), 1);
+    match &decoded[0] {
+        Mutation::SetOverlayTokens(actual) => {
+            assert_eq!(*actual, tokens);
+            assert_eq!(actual.sheet_corner_radius, 24);
+            assert_eq!(actual.dialog_corner_radius, 16);
+            assert_eq!(actual.popup_corner_radius, 12);
+            assert_eq!(actual.sheet_elevation, 8);
+            assert_eq!(actual.dialog_elevation, 6);
+            assert_eq!(actual.popup_elevation, 4);
+            assert_eq!(actual.sheet_padding, 20);
+            assert_eq!(actual.dialog_padding, 16);
+            assert_eq!(actual.popup_padding, 8);
+            assert_eq!(actual.scrim, 0x80000000);
+        }
+        other => panic!("expected SetOverlayTokens, got {other:?}"),
+    }
+}
+
+#[test]
+fn fr74_theme_builder_attaches_overlay_tokens() {
+    static TOKENS: OverlayTokens = OverlayTokens {
+        sheet_corner_radius: 24,
+        dialog_corner_radius: 16,
+        popup_corner_radius: 12,
+        sheet_elevation: 8,
+        popup_elevation: 4,
+        dialog_elevation: 6,
+        sheet_padding: 20,
+        dialog_padding: 16,
+        popup_padding: 8,
+        scrim: 0x66000000,
+    };
+
+    let theme = Theme::unified(DesignSystem::Cupertino).with_overlay_tokens(&TOKENS);
+    assert_eq!(theme.overlay, Some(&TOKENS));
+}
+
+#[test]
+fn fr74_property_kind_scrim_has_correct_wire_tag() {
+    assert_eq!(PropertyKind::Scrim as u16, 128);
+    assert_eq!(
+        PropertyKind::from_name("scrim").unwrap(),
+        PropertyKind::Scrim
+    );
+}
diff --git a/compose-rust/tests/vectors/vectors.json b/compose-rust/tests/vectors/vectors.json
index c25568d0..cf0dd8b1 100644
--- a/compose-rust/tests/vectors/vectors.json
+++ b/compose-rust/tests/vectors/vectors.json
@@ -1,6 +1,6 @@
 {
   "format": "compose-rust protocol vectors v1",
-  "schemaHashUnsigned": "6134367089496884194",
+  "schemaHashUnsigned": "2307080319921313708",
   "byteOrder": "little-endian",
   "mutations": {
     "file": "mutations.bin",
diff --git a/docs/SPEC.md b/docs/SPEC.md
index 1b501b3a..cdb09d75 100644
--- a/docs/SPEC.md
+++ b/docs/SPEC.md
@@ -4122,6 +4122,33 @@ Host API는 `ctx.show_file_dialog(FileDialog::open_files().title("...").filter("
 5. 클립보드를 쓸 수 없는 환경에서 명령은 아무 일도 하지 않고 프로세스는 계속 돕니다(시험 이름 fr73_unavailable_clipboard_is_silently_ignored).
 6. Renderer의 어느 로그 줄에도 `text`의 내용이 들어가지 않습니다(시험 이름 fr73_clipboard_text_is_never_logged).
 
+### FR-74 오버레이 테마 토큰과 기하 구조 제어 (`Draft`, 이슈 #506)
+
+요청한 쪽: darkpyonix-ember(#739, #740, `docs/DESIGN.md` §2, §4.6). Ember는 모든 OS에서 24px 연속 곡률 모서리(`corner-xl`), 지정된 팝업과 시트 그림자, 여백, 반투명 배경(scrim)을 쓰는 단일 디자인을 필요로 합니다. `Theme::with_palette`는 색상 역할만 재정의하므로 툴킷의 `Dialog`, `Menu`, `Dropdown`, `Sheet`에 대한 모양(shape), 여백(padding), 그림자/고도(elevation), 스크림(scrim)을 앱 단위로 일괄 지정할 수 없습니다. 이 항목은 애플리케이션 레벨의 오버레이 테마 토큰과 개별 노드 속성을 제공하여 기본 동작을 해치지 않고 디자인을 맞출 수 있게 합니다.
+
+- **명령 `SetOverlayTokens`(명령 태그 27)**: 오버레이 기하 구조와 효과를 지정하는 고정 길이 레코드입니다.
+  - `sheet_corner_radius: u16`: 시트 모서리 반지름(dp, 0은 시스템 기본).
+  - `dialog_corner_radius: u16`: 대화상자 모서리 반지름(dp, 0은 시스템 기본).
+  - `popup_corner_radius: u16`: 메뉴/드롭다운 모서리 반지름(dp, 0은 시스템 기본).
+  - `sheet_elevation: u16`: 시트 고도(dp, 0은 시스템 기본).
+  - `popup_elevation: u16`: 팝업 고도(dp, 0은 시스템 기본).
+  - `dialog_elevation: u16`: 대화상자 고도(dp, 0은 시스템 기본).
+  - `sheet_padding: u16`: 시트 내용 여백(dp, 0은 시스템 기본).
+  - `dialog_padding: u16`: 대화상자 내용 여백(dp, 0은 시스템 기본).
+  - `scrim: u32`: ARGB 스크림 색상(0은 시스템 기본).
+- **노드 레벨 우선순위**:
+  - `Sheet`는 노드가 자체 모양(`node.containerShape`), 자체 여백(`node.setsOwnPadding()`), 자체 배경(`node.setsOwnBackground()`), 자체 고도(`node.setsOwnElevation()`)를 지정하면 시스템 기본이나 테마 토큰보다 노드의 값을 우선 적용합니다.
+  - `PropertyKind::Scrim(128)`: 개별 `Sheet`, `Dialog`, `Menu` 노드에 지정 가능한 스크림 `Paint` 속성입니다.
+- **Host API**:
+  - `OverlayTokens`: 빌더 패턴으로 각 필드를 설정하는 `Copy` 구조체입니다.
+  - `Theme::with_overlay_tokens(self, &'static OverlayTokens) -> Self`.
+  - `Sheet`, `Dialog`, `Menu` 빌더에 `.scrim(paint)` 지원.
+
+수용 기준:
+1. `Theme::with_overlay_tokens`가 설정되면 `SetOverlayTokens`(명령 태그 27)가 인코딩되어 렌더러로 전달됩니다(시험 이름 fr74_overlay_tokens_command_round_trips).
+2. `Sheet`가 테마의 `sheet_corner_radius`와 `scrim`을 반영하여 렌더링되고, 노드에 직접 지정한 모서리 반지름과 스크림이 테마 토큰보다 우선합니다(시험 이름 fr74_sheet_honors_overlay_tokens_and_own_modifiers).
+3. `Dialog`와 `Menu`가 테마의 스크림과 모서리 설정을 반영하고, 개별 노드의 `scrim` 속성을 존중합니다(시험 이름 fr74_dialog_and_menu_honor_scrim_and_tokens).
+
 ### FR-75 텍스트 필드 타이포그래피와 아이콘 버튼 레이아웃 제어 (`Draft`, 이슈 #507)
 
 요청한 쪽: darkpyonix-ember(#739, #740, `docs/DESIGN.md` §2, §4.1). Ember NFR-L3은 여러 OS에서 동일한 컴포저와 버튼 모양을 필요로 합니다. `HostTextField`는 타이포그래피와 내부 여백을 테마 기본 규칙(`theme.rules.field`)에서 파생하고, `HostButton`은 명시적 크기를 가진 아이콘 버튼에도 테마 버튼의 최소 높이와 여백을 강제 적용하여 전역 테마 변경 시 컴포저 플레이스홀더와 플러스 버튼의 위치와 크기가 달라지는 문제가 있었습니다. 이 항목은 텍스트 필드의 명시적 타이포그래피와 내용 여백 제어, 아이콘 버튼의 크기 존중 레이아웃을 제공합니다.
diff --git a/renderer/desktop/src/protocol/Protocol.gen.kt b/renderer/desktop/src/protocol/Protocol.gen.kt
index a3a53b00..93583d78 100644
--- a/renderer/desktop/src/protocol/Protocol.gen.kt
+++ b/renderer/desktop/src/protocol/Protocol.gen.kt
@@ -8,7 +8,7 @@ import java.nio.charset.StandardCharsets
 
 enum class WidgetKind { Column, Row, Box, Text, TextField, Button, Spacer, LazyColumn, ScrollColumn, Image, Icon, Checkbox, RadioButton, Switch, Slider, ProgressIndicator, Divider, Card, Surface, Dialog, Menu, Tabs, TopAppBar, LazyRow, Tooltip, Canvas, DatePicker, TimePicker, Dropdown, Navigation, NavigationItem, Sheet, Scaffold, ScaffoldSlot, LazyGrid, FileDropTarget, ScrollRow, Chip, FloatingAction, Badge, SelectionContainer, SplitPane, CodeEditor, AbsoluteBox, Window, LinearProgressIndicator }
 
-enum class PropertyKind { Text, Placeholder, Enabled, Multiline, OnClick, OnValueChange, OnSubmit, OnFocusLost, OnKeyDown, ItemCount, ItemKey, OnRangeRequested, TypeRole, FontSize, FontWeight, LineHeight, LetterSpacing, Color, TextAlign, MaxLines, Overflow, Arrangement, Spacing, SpaceRole, Alignment, Variant, Asset, Checked, Steps, Determinate, Circular, Vertical, Open, OnDismiss, SelectedIndex, Commands, Value, Min, Max, Icon, Slot, Columns, MinColumnWidth, Spans, OnFilesEntered, OnFilesDropped, Section, Count, Collapsible, ButtonKind, Decorations, SyntaxSpans, TabWidth, OnEditRejected, OnHover, OnSave, OnDecorationClick, Folds, GutterDecorations, OnSelectionChange, Font, WordBreak, OverflowWrap, TabSize, AbsoluteSize, SoftWrap, SpanFonts, OnVisibleRangeChange, OnEditorFocusChange, OnFoldChange, OnCompositionChange, OnPointer, PointerKinds, Cursor, UserSelect, IslandZoom, WindowTitle, WindowSize, WindowMinSize, WindowResizable, WindowChrome, WindowTitleBar, WindowIcon, WindowPosition, WindowRestoreKey, WindowZoomLevel, DragFromPointer, OnWindowCloseRequested, OnWindowFocusChanged, OnWindowMoved, OnWindowDrag, OnWindowUnavailable, OnWindowClosed, OnWindowDragOver, TearOffKey, OnFocusGained, OnKeyUp, AccessibleName, TrackColor, ThumbColor, ActiveTrackColor, ContainerColor, CheckColor, ControlSize, ReportSize, Progress }
+enum class PropertyKind { Text, Placeholder, Enabled, Multiline, OnClick, OnValueChange, OnSubmit, OnFocusLost, OnKeyDown, ItemCount, ItemKey, OnRangeRequested, TypeRole, FontSize, FontWeight, LineHeight, LetterSpacing, Color, TextAlign, MaxLines, Overflow, Arrangement, Spacing, SpaceRole, Alignment, Variant, Asset, Checked, Steps, Determinate, Circular, Vertical, Open, OnDismiss, SelectedIndex, Commands, Value, Min, Max, Icon, Slot, Columns, MinColumnWidth, Spans, OnFilesEntered, OnFilesDropped, Section, Count, Collapsible, ButtonKind, Decorations, SyntaxSpans, TabWidth, OnEditRejected, OnHover, OnSave, OnDecorationClick, Folds, GutterDecorations, OnSelectionChange, Font, WordBreak, OverflowWrap, TabSize, AbsoluteSize, SoftWrap, SpanFonts, OnVisibleRangeChange, OnEditorFocusChange, OnFoldChange, OnCompositionChange, OnPointer, PointerKinds, Cursor, UserSelect, IslandZoom, WindowTitle, WindowSize, WindowMinSize, WindowResizable, WindowChrome, WindowTitleBar, WindowIcon, WindowPosition, WindowRestoreKey, WindowZoomLevel, DragFromPointer, OnWindowCloseRequested, OnWindowFocusChanged, OnWindowMoved, OnWindowDrag, OnWindowUnavailable, OnWindowClosed, OnWindowDragOver, TearOffKey, OnFocusGained, OnKeyUp, AccessibleName, TrackColor, ThumbColor, ActiveTrackColor, ContainerColor, CheckColor, ControlSize, ReportSize, Scrim, Progress }
 
 enum class Key { Enter, Other }
 
@@ -195,6 +195,20 @@ data class Theme(
 /** One role's colour in one scheme, as the application gave it. */
 data class PaletteEntry(val role: ColorRole, val dark: Boolean, val argb: Int)
 
+/** Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+data class OverlayTokens(
+    val sheetCornerRadius: Int = 0,
+    val dialogCornerRadius: Int = 0,
+    val popupCornerRadius: Int = 0,
+    val sheetElevation: Int = 0,
+    val popupElevation: Int = 0,
+    val dialogElevation: Int = 0,
+    val sheetPadding: Int = 0,
+    val dialogPadding: Int = 0,
+    val popupPadding: Int = 0,
+    val scrim: Long = 0L,
+)
+
 /**
  * What the application asked of its own window.
  *
@@ -1279,6 +1293,9 @@ sealed interface Mutation {
         val name: String,
         val filters: String,
     ) : Mutation
+
+    /** Sets application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+    data class SetOverlayTokens(val overlay: OverlayTokens) : Mutation
 }
 
 sealed interface HostEvent {
@@ -1334,7 +1351,7 @@ class ProtocolException(message: String, val offset: Int) :
     IllegalArgumentException("$message at byte offset $offset")
 
 object Protocol {
-    const val SCHEMA_HASH: Long = 6134367089496884194L
+    const val SCHEMA_HASH: Long = 2307080319921313708L
     const val PROTOCOL_VERSION: Int = 1
 
     private const val TAG_ENVELOPE = 0
@@ -1364,6 +1381,7 @@ object Protocol {
     private const val TAG_FOCUS_WINDOW = 24
     private const val TAG_REQUEST_FOCUS = 25
     private const val TAG_SHOW_FILE_DIALOG = 26
+    private const val TAG_SET_OVERLAY_TOKENS = 27
     private const val ENVELOPE_LENGTH = 12
     private const val ANIMATION_HEADER_LENGTH = 52
     private const val KEYFRAME_HEAD_LENGTH = 24
@@ -1530,6 +1548,23 @@ object Protocol {
                             readString(batch, base, available, offset + 40),
                         )
                     }
+                    TAG_SET_OVERLAY_TOKENS -> {
+                        requireRecordLength(length, 28, offset)
+                        Mutation.SetOverlayTokens(
+                            OverlayTokens(
+                                sheetCornerRadius = readU16(batch, base, available, offset + 4),
+                                dialogCornerRadius = readU16(batch, base, available, offset + 6),
+                                popupCornerRadius = readU16(batch, base, available, offset + 8),
+                                sheetElevation = readU16(batch, base, available, offset + 10),
+                                popupElevation = readU16(batch, base, available, offset + 12),
+                                dialogElevation = readU16(batch, base, available, offset + 14),
+                                sheetPadding = readU16(batch, base, available, offset + 16),
+                                dialogPadding = readU16(batch, base, available, offset + 18),
+                                popupPadding = readU16(batch, base, available, offset + 20),
+                                scrim = readU32(batch, base, available, offset + 24),
+                            ),
+                        )
+                    }
                     TAG_SET_WINDOW -> {
                         requireRecordLength(length, 32, offset)
                         // Bit 0 is `resizable`, bit 1 asks for every whole-dp size change; the
@@ -2283,6 +2318,7 @@ object Protocol {
         125 -> PropertyKind.CheckColor
         126 -> PropertyKind.ControlSize
         127 -> PropertyKind.ReportSize
+        128 -> PropertyKind.Scrim
         27 -> PropertyKind.Progress
         else -> throw ProtocolException("unknown property tag $tag", offset)
     }
diff --git a/renderer/desktop/src/renderer/ComposeRustHost.kt b/renderer/desktop/src/renderer/ComposeRustHost.kt
index 03c40c11..681dcea3 100644
--- a/renderer/desktop/src/renderer/ComposeRustHost.kt
+++ b/renderer/desktop/src/renderer/ComposeRustHost.kt
@@ -484,6 +484,7 @@ private fun ComposeRustTree(
         chainOf = { asset -> (host.table.assets.asset(asset) as? Asset.Font)?.chain },
         brushOf = { asset -> (host.table.assets.asset(asset) as? Asset.Brush)?.brush },
         windowBackdrop = asked && platformBacksWindowWithMaterial(),
+        overlay = host.table.overlayTokens,
     )
     // The animations the Host handed over are played here, on this side's frame clock, and
     // the Host is not called while they play unless one asked for events. The loop waits
diff --git a/renderer/desktop/src/renderer/DesignSystem.kt b/renderer/desktop/src/renderer/DesignSystem.kt
index 682b9586..f603e80f 100644
--- a/renderer/desktop/src/renderer/DesignSystem.kt
+++ b/renderer/desktop/src/renderer/DesignSystem.kt
@@ -22,6 +22,7 @@ import dev.darkpyonix.composerust.protocol.DesignTokens
 import androidx.compose.animation.core.FiniteAnimationSpec
 import dev.darkpyonix.composerust.protocol.MaterialRole
 import dev.darkpyonix.composerust.protocol.MotionRole
+import dev.darkpyonix.composerust.protocol.OverlayTokens
 import dev.darkpyonix.composerust.protocol.Paint
 import dev.darkpyonix.composerust.protocol.ShapeRole
 import dev.darkpyonix.composerust.protocol.SpaceRole
@@ -180,6 +181,11 @@ class ResolvedTheme(
      * given is enough to find the rest of the stack: see [chainFor].
      */
     val fontChains: List<FontChain> = emptyList(),
+
+    /**
+     * Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet.
+     */
+    val overlay: OverlayTokens? = null,
 ) {
     /**
      * The stack [family] is the first family of, or null where it is not one.
@@ -1638,6 +1644,8 @@ fun resolveTheme(
      * family lacks in the families after it.
      */
     chainOf: (Int) -> FontChain? = { null },
+    /** Application-level overlay theme tokens for Dialog, Menu, Dropdown and Sheet. */
+    overlay: OverlayTokens? = null,
 ): ResolvedTheme {
     val system = when {
         theme == null -> adaptiveSystem(platform, DesignSystem.Material3)
@@ -1684,6 +1692,7 @@ fun resolveTheme(
         platform,
         application,
         chains,
+        overlay,
     )
 }
 
diff --git a/renderer/desktop/src/renderer/HostContainer.kt b/renderer/desktop/src/renderer/HostContainer.kt
index 9914be33..bafe5709 100644
--- a/renderer/desktop/src/renderer/HostContainer.kt
+++ b/renderer/desktop/src/renderer/HostContainer.kt
@@ -7,6 +7,7 @@ import dev.darkpyonix.composerust.protocol.TypeRole
 import dev.darkpyonix.composerust.ui.textStyle
 import androidx.compose.foundation.background
 import androidx.compose.foundation.border
+import androidx.compose.foundation.shape.RoundedCornerShape
 import androidx.compose.foundation.layout.Arrangement
 import androidx.compose.foundation.layout.Box
 import androidx.compose.foundation.layout.Column
@@ -45,12 +46,20 @@ import dev.darkpyonix.composerust.ui.resolvedShape
 import dev.darkpyonix.composerust.ui.weightOf
 import dev.darkpyonix.composerust.design.liftsOffThePage
 
+/** True when the Host set a shape modifier of its own, which replaces the default shape. */
+internal fun Node.setsOwnShape(): Boolean =
+    modifiers.any {
+        it is ProtocolModifier.Shape ||
+            it is ProtocolModifier.ShapeRole ||
+            it is ProtocolModifier.CornerEach
+    }
+
 /**
  * The corner this container is drawn with: the one the Host asked for if it asked, and the
  * design system's corner for this kind of container otherwise.
  */
 internal fun Node.containerShape(style: ContainerStyle, theme: ResolvedTheme): Shape =
-    if (modifiers.any { it is ProtocolModifier.Shape || it is ProtocolModifier.ShapeRole || it is ProtocolModifier.CornerEach }) {
+    if (setsOwnShape()) {
         modifiers.resolvedShape(theme)
     } else {
         style.shape
@@ -98,12 +107,34 @@ internal fun Modifier.containerDecoration(
     theme: ResolvedTheme,
     /** True for a menu, a dialog or a tooltip: a sheet over the page rather than in it. */
     overlay: Boolean = false,
+    role: ContainerRole? = null,
 ): Modifier {
-    val shape = node.containerShape(style, theme)
-    val raised = if (node.setsOwnElevation()) {
+    val shape = if (node.setsOwnShape()) {
+        node.containerShape(style, theme)
+    } else if (overlay) {
+        when (role) {
+            ContainerRole.Dialog -> theme.overlay?.dialogCornerRadius?.takeIf { it > 0 }?.let { RoundedCornerShape(it.dp) }
+            ContainerRole.Menu -> theme.overlay?.popupCornerRadius?.takeIf { it > 0 }?.let { RoundedCornerShape(it.dp) }
+            else -> null
+        } ?: style.shape
+    } else {
+        style.shape
+    }
+    val effectiveElevation = if (node.setsOwnElevation()) {
+        0.dp
+    } else if (overlay) {
+        when (role) {
+            ContainerRole.Dialog -> theme.overlay?.dialogElevation?.takeIf { it > 0 }?.let { it.dp }
+            ContainerRole.Menu -> theme.overlay?.popupElevation?.takeIf { it > 0 }?.let { it.dp }
+            else -> null
+        } ?: style.elevation
+    } else {
+        style.elevation
+    }
+    val raised = if (node.setsOwnElevation() || effectiveElevation.value <= 0f) {
         this
     } else {
-        theme.rules.elevation(this, style.elevation, shape, theme)
+        theme.rules.elevation(this, effectiveElevation, shape, theme)
     }
     val material = style.material
     // A Host that named its own Background has already had it applied, so painting the
@@ -139,6 +170,17 @@ internal fun Modifier.containerDecoration(
         .then(
             if (node.setsOwnPadding()) {
                 Modifier
+            } else if (overlay) {
+                val customPadding = when (role) {
+                    ContainerRole.Dialog -> theme.overlay?.dialogPadding?.takeIf { it > 0 }?.let { it.dp }
+                    ContainerRole.Menu -> theme.overlay?.popupPadding?.takeIf { it > 0 }?.let { it.dp }
+                    else -> null
+                }
+                if (customPadding != null) {
+                    Modifier.padding(customPadding)
+                } else {
+                    Modifier.padding(horizontal = style.horizontalPadding, vertical = style.verticalPadding)
+                }
             } else {
                 Modifier.padding(horizontal = style.horizontalPadding, vertical = style.verticalPadding)
             },
diff --git a/renderer/desktop/src/renderer/HostDropdown.kt b/renderer/desktop/src/renderer/HostDropdown.kt
index 52b3f526..dc1fcdd9 100644
--- a/renderer/desktop/src/renderer/HostDropdown.kt
+++ b/renderer/desktop/src/renderer/HostDropdown.kt
@@ -196,6 +196,29 @@ private fun OpeningList(
             )
         }
         if (open) {
+            val popupShape = if (node.setsOwnShape()) {
+                node.containerShape(style, theme)
+            } else {
+                theme.overlay?.popupCornerRadius?.takeIf { it > 0 }?.let { androidx.compose.foundation.shape.RoundedCornerShape(it.dp) } ?: style.shape
+            }
+            val popupElevation = if (node.setsOwnElevation()) {
+                0.dp
+            } else {
+                theme.overlay?.popupElevation?.takeIf { it > 0 }?.let { it.dp } ?: style.elevation
+            }
+            val popupRaised = if (node.setsOwnElevation() || popupElevation.value <= 0f) {
+                Modifier
+            } else {
+                theme.rules.elevation(Modifier, popupElevation, popupShape, theme)
+            }
+            val explicitPopupPad = theme.overlay?.popupPadding
+            val popupPadding = if (node.setsOwnPadding()) {
+                Modifier
+            } else if (explicitPopupPad != null && explicitPopupPad > 0) {
+                Modifier.padding(explicitPopupPad.dp)
+            } else {
+                Modifier.padding(vertical = theme.space(SpaceRole.Xs))
+            }
             Popup(
                 popupPositionProvider = BelowAnchorPositionProvider,
                 onDismissRequest = { open = false },
@@ -205,10 +228,11 @@ private fun OpeningList(
                     Column(
                         Modifier
                             .testTag(dropdownPopupTestTag(node.id))
-                            .clip(style.shape)
-                            .background(style.container, style.shape)
-                            .border(style.borderWidth, style.borderColor, style.shape)
-                            .padding(vertical = theme.space(SpaceRole.Xs)),
+                            .then(popupRaised)
+                            .clip(popupShape)
+                            .background(style.container, popupShape)
+                            .border(style.borderWidth, style.borderColor, popupShape)
+                            .then(popupPadding),
                     ) {
                         options.forEachIndexed { index, childId ->
                             key(childId) {
diff --git a/renderer/desktop/src/renderer/HostOverlay.kt b/renderer/desktop/src/renderer/HostOverlay.kt
index 43483ea3..834c0246 100644
--- a/renderer/desktop/src/renderer/HostOverlay.kt
+++ b/renderer/desktop/src/renderer/HostOverlay.kt
@@ -50,6 +50,7 @@ import dev.darkpyonix.composerust.ui.node.Node
 import dev.darkpyonix.composerust.ui.node.NodeTable
 import dev.darkpyonix.composerust.ui.node.RenderNode
 import dev.darkpyonix.composerust.ui.node.nodeTestTag
+import dev.darkpyonix.composerust.ui.paintProp
 import dev.darkpyonix.composerust.ui.textStyle
 import dev.darkpyonix.composerust.design.LocalGlassBackdrop
 import androidx.compose.runtime.CompositionLocalProvider
@@ -194,6 +195,9 @@ internal fun HostDialog(
     val open = rememberOverlayOpen(node)
     if (!open.value) return
     val style = theme.rules.container(ContainerRole.Dialog, theme)
+    val scrim = node.paintProp(PropertyKind.Scrim)?.let { theme.color(it) }
+        ?: theme.overlay?.scrim?.takeIf { it != 0L }?.let { androidx.compose.ui.graphics.Color(it.toULong()) }
+        ?: style.scrim
     val surfaceRequester = remember(node.id) { FocusRequester() }
     val onEscape = { dismiss(node, dispatcher) }
     Popup(
@@ -206,14 +210,13 @@ internal fun HostDialog(
     ) {
         ModalOverlayFocus(node.id, surfaceRequester, onEscape)
         Box(
-            Modifier.fillMaxSize().background(style.scrim),
+            Modifier.fillMaxSize().background(scrim).testTag(dialogTestTag(node.id)),
             contentAlignment = Alignment.Center,
         ) {
             Column(
                 modifier
-                    .testTag(dialogTestTag(node.id))
                     .modalOverlaySurface(node.id, surfaceRequester)
-                    .containerDecoration(node, style, theme, overlay = true),
+                    .containerDecoration(node, style, theme, overlay = true, role = ContainerRole.Dialog),
             ) {
                 node.children.forEach { childId ->
                     key(childId) { RenderNode(childId, table, dispatcher) }
@@ -265,7 +268,7 @@ internal fun HostMenu(
                 Column(
                     Modifier
                         .testTag(menuPopupTestTag(node.id))
-                        .containerDecoration(node, style, theme, overlay = true),
+                        .containerDecoration(node, style, theme, overlay = true, role = ContainerRole.Menu),
                 ) {
                     entries.forEach { childId ->
                         key(childId) { RenderNode(childId, table, dispatcher) }
diff --git a/renderer/desktop/src/renderer/HostSheet.kt b/renderer/desktop/src/renderer/HostSheet.kt
index f5a12bb9..74cd7dce 100644
--- a/renderer/desktop/src/renderer/HostSheet.kt
+++ b/renderer/desktop/src/renderer/HostSheet.kt
@@ -34,6 +34,7 @@ import androidx.compose.ui.window.Popup
 import androidx.compose.ui.window.PopupProperties
 import dev.darkpyonix.composerust.design.ResolvedTheme
 import dev.darkpyonix.composerust.design.SheetEdge
+import dev.darkpyonix.composerust.protocol.PropertyKind
 import dev.darkpyonix.composerust.protocol.ShapeRole
 import dev.darkpyonix.composerust.runtime.EventDispatcher
 import dev.darkpyonix.composerust.runtime.LocalWindowSizeClass
@@ -41,6 +42,7 @@ import dev.darkpyonix.composerust.ui.node.Node
 import dev.darkpyonix.composerust.ui.node.NodeTable
 import dev.darkpyonix.composerust.ui.node.RenderNode
 import dev.darkpyonix.composerust.ui.node.nodeTestTag
+import dev.darkpyonix.composerust.ui.paintProp
 import kotlin.math.roundToInt
 
 /** How far a sheet has to be pulled towards its edge before letting go closes it. */
@@ -84,6 +86,65 @@ internal fun HostSheet(
     val surfaceRequester = remember(node.id) { FocusRequester() }
     val onEscape = { dismiss(node, dispatcher) }
 
+    val scrim = node.paintProp(PropertyKind.Scrim)?.let { theme.color(it) }
+        ?: theme.overlay?.scrim?.takeIf { it != 0L }?.let { androidx.compose.ui.graphics.Color(it.toULong()) }
+        ?: style.scrim
+
+    val shape = if (node.setsOwnShape()) {
+        node.containerShape(
+            dev.darkpyonix.composerust.design.ContainerStyle(
+                container = style.container,
+                content = androidx.compose.ui.graphics.Color.Unspecified,
+                shape = style.shape,
+                elevation = style.elevation,
+                borderWidth = style.borderWidth,
+                borderColor = style.borderColor,
+                horizontalPadding = 0.dp,
+                verticalPadding = 0.dp,
+                separator = null,
+                scrim = style.scrim,
+                typeRole = dev.darkpyonix.composerust.protocol.TypeRole.Body,
+            ),
+            theme,
+        )
+    } else {
+        theme.overlay?.sheetCornerRadius?.takeIf { it > 0 }?.let {
+            if (bottom) {
+                androidx.compose.foundation.shape.RoundedCornerShape(it.dp, it.dp, 0.dp, 0.dp)
+            } else {
+                androidx.compose.foundation.shape.RoundedCornerShape(it.dp, 0.dp, 0.dp, it.dp)
+            }
+        } ?: style.shape
+    }
+
+    val sheetElevation = if (node.setsOwnElevation()) {
+        0.dp
+    } else {
+        theme.overlay?.sheetElevation?.takeIf { it > 0 }?.let { it.dp } ?: style.elevation
+    }
+    val raised = if (node.setsOwnElevation() || sheetElevation.value <= 0f) {
+        Modifier
+    } else {
+        theme.rules.elevation(Modifier, sheetElevation, shape, theme)
+    }
+
+    val containerBg = if (node.setsOwnBackground()) Modifier else Modifier.background(style.container)
+
+    val outline = if (style.borderWidth.value > 0f) {
+        Modifier.border(style.borderWidth, style.borderColor, shape)
+    } else {
+        Modifier
+    }
+
+    val explicitSheetPad = theme.overlay?.sheetPadding
+    val sheetPadding = if (node.setsOwnPadding()) {
+        Modifier
+    } else if (explicitSheetPad != null && explicitSheetPad > 0) {
+        Modifier.padding(explicitSheetPad.dp)
+    } else {
+        Modifier.padding(style.padding)
+    }
+
     Popup(
         alignment = Alignment.Center,
         properties = PopupProperties(focusable = true, dismissOnBackPress = false),
@@ -94,7 +155,7 @@ internal fun HostSheet(
             Modifier
                 .testTag(sheetScrimTestTag(node.id))
                 .fillMaxSize()
-                .background(style.scrim)
+                .background(scrim)
                 // Pressing what the sheet covers is the other way of asking to close it,
                 // and it gets the same answer: one dismissal. A raw tap detector rather
                 // than `clickable`, because a scrim is not a button and saying it is one
@@ -107,11 +168,6 @@ internal fun HostSheet(
             } else {
                 Modifier.fillMaxHeight().fillMaxWidth(style.widthFraction)
             }
-            val outline = if (style.borderWidth.value > 0f) {
-                Modifier.border(style.borderWidth, style.borderColor, style.shape)
-            } else {
-                Modifier
-            }
             // Two boxes rather than one: the outer is the sheet itself, which is what is
             // dragged and what the design system sized, and the inner carries whatever
             // Modifiers the Host put on the node, which belong inside its surface.
@@ -127,8 +183,9 @@ internal fun HostSheet(
                             IntOffset(travel.roundToInt(), 0)
                         }
                     }
-                    .clip(style.shape)
-                    .background(style.container)
+                    .then(raised)
+                    .clip(shape)
+                    .then(containerBg)
                     .then(outline)
                     .draggable(
                         orientation = if (bottom) Orientation.Vertical else Orientation.Horizontal,
@@ -143,7 +200,7 @@ internal fun HostSheet(
                     // swallowed here rather than reaching the scrim behind.
                     .pointerInput(Unit) { detectTapGestures { } },
             ) {
-                Column(modifier.fillMaxWidth().padding(style.padding)) {
+                Column(modifier.fillMaxWidth().then(sheetPadding)) {
                     style.handle?.let { handle ->
                         Box(Modifier.fillMaxWidth(), contentAlignment = Alignment.Center) {
                             Box(
diff --git a/renderer/desktop/src/renderer/NodeTable.kt b/renderer/desktop/src/renderer/NodeTable.kt
index 1d4dab3f..e74429f5 100644
--- a/renderer/desktop/src/renderer/NodeTable.kt
+++ b/renderer/desktop/src/renderer/NodeTable.kt
@@ -8,6 +8,7 @@ import androidx.compose.runtime.setValue
 import dev.darkpyonix.composerust.foundation.ChromeInsets
 import dev.darkpyonix.composerust.foundation.MessageQueue
 import dev.darkpyonix.composerust.protocol.Mutation
+import dev.darkpyonix.composerust.protocol.OverlayTokens
 import dev.darkpyonix.composerust.protocol.PropertyKind
 import dev.darkpyonix.composerust.protocol.PropertyValue
 import dev.darkpyonix.composerust.protocol.Theme
@@ -187,6 +188,10 @@ class NodeTable {
     var theme: Theme? by mutableStateOf(null)
         private set
 
+    /** Null until the Host sends an overlay tokens record. */
+    var overlayTokens: OverlayTokens? by mutableStateOf(null)
+        private set
+
     /** Null until the Host sends its first `SetWindow` record. */
     var window: Window? by mutableStateOf(null)
         private set
@@ -320,6 +325,7 @@ class NodeTable {
                     fail(TableError.INVALID_PALETTE, "theme palette: $problem")
                 }
             }
+            is Mutation.SetOverlayTokens -> overlayTokens = mutation.overlay
             // Read before there is a window to apply it to. The platform layer asks for
             // this out of the first batch and builds the window from it, so by the time
             // the batch is applied the window already looks the way it says. Keeping it
@@ -974,6 +980,11 @@ class NodeTable {
                     widget == WidgetKind.Menu ||
                     // A sheet is an overlay on the same terms as a dialog.
                     widget == WidgetKind.Sheet
+
+                // A custom scrim colour for modal and popup overlays.
+                PropertyKind.Scrim -> widget == WidgetKind.Dialog ||
+                    widget == WidgetKind.Menu ||
+                    widget == WidgetKind.Sheet
                 // A Dropdown's selection is a position in its own children, the same thing
                 // a tab strip's selection is.
                 PropertyKind.SelectedIndex ->
diff --git a/renderer/desktop/test/HostOverlayTokensTest.kt b/renderer/desktop/test/HostOverlayTokensTest.kt
new file mode 100644
index 00000000..49dfd4df
--- /dev/null
+++ b/renderer/desktop/test/HostOverlayTokensTest.kt
@@ -0,0 +1,149 @@
+package dev.darkpyonix.composerust.test
+
+import androidx.compose.foundation.layout.requiredSize
+import androidx.compose.runtime.CompositionLocalProvider
+import androidx.compose.ui.Modifier
+import androidx.compose.ui.platform.LocalDensity
+import androidx.compose.ui.test.ExperimentalTestApi
+import androidx.compose.ui.test.assertTextEquals
+import androidx.compose.ui.test.onNodeWithTag
+import androidx.compose.ui.test.runComposeUiTest
+import androidx.compose.ui.unit.Density
+import androidx.compose.ui.unit.dp
+import dev.darkpyonix.composerust.foundation.dialogTestTag
+import dev.darkpyonix.composerust.foundation.sheetTestTag
+import dev.darkpyonix.composerust.protocol.ColorScheme
+import dev.darkpyonix.composerust.protocol.DesignSystem
+import dev.darkpyonix.composerust.protocol.Mutation
+import dev.darkpyonix.composerust.protocol.OverlayTokens
+import dev.darkpyonix.composerust.protocol.Paint
+import dev.darkpyonix.composerust.protocol.PropertyKind
+import dev.darkpyonix.composerust.protocol.PropertyValue
+import dev.darkpyonix.composerust.protocol.Theme
+import dev.darkpyonix.composerust.protocol.WidgetKind
+import dev.darkpyonix.composerust.runtime.ComposeRustContent
+import dev.darkpyonix.composerust.runtime.ComposeRustHost
+import dev.darkpyonix.composerust.runtime.rememberComposeRustHost
+import dev.darkpyonix.composerust.tooling.FakeHostConnection
+import dev.darkpyonix.composerust.ui.node.nodeTestTag
+import dev.darkpyonix.composerust.ui.paintProp
+import dev.darkpyonix.composerust.ui.platform.FrameRequestSource
+import dev.darkpyonix.composerust.ui.platform.LocalFrameRequests
+import kotlin.test.Test
+import kotlin.test.assertEquals
+import kotlin.test.assertNotNull
+
+@OptIn(ExperimentalTestApi::class)
+class HostOverlayTokensTest {
+    private val frames = FrameRequestSource()
+
+    @Test
+    fun fr74_sheet_honors_overlay_tokens_and_own_modifiers() = runComposeUiTest {
+        val sheetId = 1
+        val textId = 2
+        val tokens = OverlayTokens(
+            sheetCornerRadius = 24,
+            sheetElevation = 8,
+            sheetPadding = 20,
+            scrim = 0x80000000L,
+        )
+        val connection = FakeHostConnection(
+            listOf(
+                Mutation.SetTheme(
+                    Theme(DesignSystem.Material3, DesignSystem.Material3, ColorScheme.Light, adaptive = false),
+                ),
+                Mutation.SetOverlayTokens(tokens),
+                Mutation.Create(sheetId, WidgetKind.Sheet),
+                Mutation.SetProp(sheetId, PropertyKind.Open, PropertyValue.Bool(true)),
+                Mutation.Create(textId, WidgetKind.Text),
+                Mutation.SetProp(textId, PropertyKind.Text, PropertyValue.Text("Settings")),
+                Mutation.Insert(sheetId, textId, 0),
+            ),
+        )
+        lateinit var host: ComposeRustHost
+        setContent {
+            CompositionLocalProvider(
+                LocalFrameRequests provides frames,
+                LocalDensity provides Density(1f),
+            ) {
+                host = rememberComposeRustHost(connection)
+                ComposeRustContent(
+                    host,
+                    Modifier.requiredSize(500.dp, 800.dp),
+                )
+            }
+        }
+        waitForIdle()
+
+        assertEquals(tokens, host.table.overlayTokens)
+        onNodeWithTag(nodeTestTag(textId)).assertTextEquals("Settings")
+
+        // Overriding scrim on the node directly
+        val scrimBits = (2L shl 32) or (0x40112233L and 0xFFFFFFFFL)
+        connection.scheduleFrame(
+            listOf(
+                Mutation.SetProp(sheetId, PropertyKind.Scrim, PropertyValue.Integer(scrimBits)),
+            ),
+        )
+        frames.request()
+        waitForIdle()
+
+        val node = host.table.node(sheetId)
+        assertNotNull(node)
+        assertEquals(Paint.Literal(0x40112233), node.paintProp(PropertyKind.Scrim))
+    }
+
+    @Test
+    fun fr74_dialog_and_menu_honor_scrim_and_tokens() = runComposeUiTest {
+        val root = 10
+        val dialogId = 11
+        val dialogText = 12
+        val tokens = OverlayTokens(
+            dialogCornerRadius = 16,
+            popupCornerRadius = 12,
+            dialogElevation = 6,
+            popupElevation = 4,
+            dialogPadding = 16,
+            popupPadding = 8,
+            scrim = 0x66000000L,
+        )
+        val scrimBits = (2L shl 32) or (0x80000000L and 0xFFFFFFFFL)
+        val connection = FakeHostConnection(
+            listOf(
+                Mutation.SetTheme(
+                    Theme(DesignSystem.Cupertino, DesignSystem.Cupertino, ColorScheme.Light, adaptive = false),
+                ),
+                Mutation.SetOverlayTokens(tokens),
+                Mutation.Create(root, WidgetKind.Column),
+                Mutation.Create(dialogId, WidgetKind.Dialog),
+                Mutation.SetProp(dialogId, PropertyKind.Open, PropertyValue.Bool(true)),
+                Mutation.SetProp(dialogId, PropertyKind.Scrim, PropertyValue.Integer(scrimBits)),
+                Mutation.Create(dialogText, WidgetKind.Text),
+                Mutation.SetProp(dialogText, PropertyKind.Text, PropertyValue.Text("Confirm")),
+                Mutation.Insert(dialogId, dialogText, 0),
+                Mutation.Insert(root, dialogId, 0),
+            ),
+        )
+        lateinit var host: ComposeRustHost
+        setContent {
+            CompositionLocalProvider(
+                LocalFrameRequests provides frames,
+                LocalDensity provides Density(1f),
+            ) {
+                host = rememberComposeRustHost(connection)
+                ComposeRustContent(
+                    host,
+                    Modifier.requiredSize(800.dp, 600.dp),
+                )
+            }
+        }
+        waitForIdle()
+
+        assertEquals(tokens, host.table.overlayTokens)
+        onNodeWithTag(nodeTestTag(dialogText)).assertTextEquals("Confirm")
+        onNodeWithTag(dialogTestTag(dialogId)).assertExists()
+        val dialogNode = host.table.node(dialogId)
+        assertNotNull(dialogNode)
+        assertEquals(Paint.Literal(0x80000000.toInt()), dialogNode.paintProp(PropertyKind.Scrim))
+    }
+}
diff --git a/renderer/desktop/test/ProtocolVectorsTest.kt b/renderer/desktop/test/ProtocolVectorsTest.kt
index 98f70f56..1418fa11 100644
--- a/renderer/desktop/test/ProtocolVectorsTest.kt
+++ b/renderer/desktop/test/ProtocolVectorsTest.kt
@@ -101,6 +101,7 @@ class ProtocolVectorsTest {
                 is Mutation.AppendText -> mutation.nodeId !in created
                 // The theme applies to the tree, not to a node, so it is never a bad record.
                 is Mutation.SetTheme -> false
+                is Mutation.SetOverlayTokens -> false
                 // Nor does the window: it is about the frame around the tree.
                 is Mutation.SetWindow -> false
                 // A window's theme and bringing one forward name a node the vector created.