diff --git a/package.json b/package.json
index 83c3c23b..04963fdf 100644
--- a/package.json
+++ b/package.json
@@ -105,6 +105,7 @@
     "vitest": "^0.34.4"
   },
   "dependencies": {
+    "@radix-ui/react-context-menu": "^2.1.5",
     "@radix-ui/react-dropdown-menu": "^2.0.6",
     "@radix-ui/react-form": "^0.0.3",
     "@radix-ui/react-separator": "^1.0.3",
diff --git a/src/components/Menu/ContextMenu.stories.tsx b/src/components/Menu/ContextMenu.stories.tsx
new file mode 100644
index 00000000..b51a20d2
--- /dev/null
+++ b/src/components/Menu/ContextMenu.stories.tsx
@@ -0,0 +1,80 @@
+/*
+Copyright 2023 New Vector Ltd
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+import React from "react";
+import { Meta, StoryFn } from "@storybook/react";
+import UserProfileIcon from "@vector-im/compound-design-tokens/icons/user-profile.svg";
+import NotificationsIcon from "@vector-im/compound-design-tokens/icons/notifications.svg";
+import ChatProblemIcon from "@vector-im/compound-design-tokens/icons/chat-problem.svg";
+import LeaveIcon from "@vector-im/compound-design-tokens/icons/leave.svg";
+
+import { ContextMenu as ContextMenuComponent } from "./ContextMenu";
+import { MenuItem } from "./MenuItem";
+import { Separator } from "../Separator/Separator";
+
+export default {
+  title: "Menu/ContextMenu",
+  component: ContextMenuComponent,
+  tags: ["autodocs"],
+  argTypes: {},
+  args: {},
+} as Meta<typeof ContextMenuComponent>;
+
+const Template: StoryFn<typeof ContextMenuComponent> = (args) => {
+  return (
+    <ContextMenuComponent
+      {...args}
+      title="Settings"
+      trigger={
+        <div
+          style={{
+            borderRadius: 24,
+            background: "var(--cpd-color-bg-subtle-secondary)",
+            inlineSize: 300,
+            blockSize: 200,
+            textAlign: "center",
+            display: "flex",
+            alignItems: "center",
+            justifyContent: "center",
+            padding: 32,
+            boxSizing: "border-box",
+          }}
+        >
+          Right click or long press to open menu
+        </div>
+      }
+      hasAccessibleAlternative={false}
+    >
+      <MenuItem Icon={UserProfileIcon} label="Profile" onSelect={() => {}} />
+      <MenuItem
+        Icon={NotificationsIcon}
+        label="Notifications"
+        onSelect={() => {}}
+      />
+      <MenuItem Icon={ChatProblemIcon} label="Feedback" onSelect={() => {}} />
+      <Separator />
+      <MenuItem
+        kind="critical"
+        Icon={LeaveIcon}
+        label="Sign out"
+        onSelect={() => {}}
+      />
+    </ContextMenuComponent>
+  );
+};
+
+export const ContextMenu = Template.bind({});
+ContextMenu.args = {};
diff --git a/src/components/Menu/ContextMenu.test.tsx b/src/components/Menu/ContextMenu.test.tsx
new file mode 100644
index 00000000..aa75614f
--- /dev/null
+++ b/src/components/Menu/ContextMenu.test.tsx
@@ -0,0 +1,50 @@
+/*
+Copyright 2023 New Vector Ltd
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+import { describe, it, expect, vi } from "vitest";
+import { render, screen } from "@testing-library/react";
+import React from "react";
+import UserProfileIcon from "@vector-im/compound-design-tokens/icons/user-profile.svg";
+
+import { ContextMenu } from "./ContextMenu";
+import { MenuItem } from "./MenuItem";
+import userEvent from "@testing-library/user-event";
+
+describe("ContextMenu", () => {
+  it("opens by right-clicking", async () => {
+    const onOpenChange = vi.fn();
+    render(
+      <ContextMenu
+        title="Settings"
+        onOpenChange={onOpenChange}
+        trigger={<div>Open menu</div>}
+        hasAccessibleAlternative
+      >
+        <MenuItem Icon={UserProfileIcon} label="Profile" onSelect={() => {}} />
+      </ContextMenu>,
+    );
+
+    expect(screen.queryByRole("menu")).toBe(null);
+    // Right-click the trigger area
+    const trigger = screen.getByText("Open menu");
+    await userEvent.pointer([{ target: trigger }, { keys: "[MouseRight]" }]);
+    expect(onOpenChange).toHaveBeenLastCalledWith(true);
+  });
+
+  // Here it would be nice to also test opening by long-pressing, but that
+  // requires fake timers, and user-event appears to stall out when using fake
+  // timers :(
+});
diff --git a/src/components/Menu/ContextMenu.tsx b/src/components/Menu/ContextMenu.tsx
new file mode 100644
index 00000000..bad853fe
--- /dev/null
+++ b/src/components/Menu/ContextMenu.tsx
@@ -0,0 +1,142 @@
+/*
+Copyright 2023 New Vector Ltd
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+import React, { FC, ReactNode, useCallback, useMemo, useState } from "react";
+import {
+  Root,
+  Trigger,
+  Portal,
+  Content,
+  ContextMenuItem,
+} from "@radix-ui/react-context-menu";
+import { FloatingMenu } from "./FloatingMenu";
+import { Drawer } from "vaul";
+import classnames from "classnames";
+import drawerStyles from "./DrawerMenu.module.css";
+import { MenuContext, MenuData, MenuItemWrapperProps } from "./MenuContext";
+import { DrawerMenu } from "./DrawerMenu";
+import { getPlatform } from "../../utils/platform";
+
+interface Props {
+  /**
+   * The menu title.
+   */
+  title: string;
+  /**
+   * Event handler called when the open state of the menu changes.
+   */
+  onOpenChange?: (open: boolean) => void;
+  /**
+   * The trigger that can be right-clicked or long-pressed to open the menu.
+   * This must be a component that accepts a ref and spreads props.
+   * https://www.radix-ui.com/primitives/docs/guides/composition
+   */
+  trigger: ReactNode;
+  /**
+   * Whether the functionality of this menu is available through some other
+   * keyboard-accessible means. Preferably this should be true, because context
+   * menus are potentially difficult to discover, but if false the trigger will
+   * become focusable so that it can be opened via keyboard navigation.
+   */
+  hasAccessibleAlternative: boolean;
+  /**
+   * The menu contents.
+   */
+  children: ReactNode;
+}
+
+const ContextMenuItemWrapper: FC<MenuItemWrapperProps> = ({
+  onSelect,
+  children,
+}) => (
+  <ContextMenuItem onSelect={onSelect ?? undefined} asChild>
+    {children}
+  </ContextMenuItem>
+);
+
+/**
+ * A menu opened by right-clicking or long-pressing another UI element.
+ */
+export const ContextMenu: FC<Props> = ({
+  title,
+  onOpenChange: onOpenChangeProp,
+  trigger: triggerProp,
+  hasAccessibleAlternative,
+  children: childrenProp,
+}) => {
+  const [open, setOpen] = useState(false);
+  const onOpenChange = useCallback(
+    (value: boolean) => {
+      setOpen(value);
+      onOpenChangeProp?.(value);
+    },
+    [setOpen, onOpenChangeProp],
+  );
+
+  // Normally, the menu takes the form of a floating box. But on Android and
+  // iOS, the menu should morph into a drawer
+  const platform = getPlatform();
+  const drawer = platform === "android" || platform === "ios";
+  const context: MenuData = useMemo(
+    () => ({
+      MenuItemWrapper: drawer ? null : ContextMenuItemWrapper,
+      onOpenChange,
+    }),
+    [onOpenChange],
+  );
+  const children = (
+    <MenuContext.Provider value={context}>{childrenProp}</MenuContext.Provider>
+  );
+
+  const trigger = (
+    <Trigger
+      aria-haspopup="menu"
+      tabIndex={hasAccessibleAlternative ? undefined : 0}
+      asChild
+    >
+      {triggerProp}
+    </Trigger>
+  );
+
+  // This is a small hack: Vaul drawers only support buttons as triggers, so
+  // we end up mounting an empty Radix context menu tree alongside the
+  // drawer tree, purely so we can use its Trigger component (which supports
+  // touch for free). The resulting behavior and DOM tree looks exactly the
+  // same as if Vaul provided a long-press trigger of its own, so I think
+  // this is fine.
+  return drawer ? (
+    <>
+      <Root onOpenChange={onOpenChange}>{trigger}</Root>
+      <Drawer.Root open={open} onOpenChange={onOpenChange}>
+        <Drawer.Portal>
+          <Drawer.Overlay className={classnames(drawerStyles.bg)} />
+          <Drawer.Content asChild>
+            <DrawerMenu title={title}>{children}</DrawerMenu>
+          </Drawer.Content>
+        </Drawer.Portal>
+      </Drawer.Root>
+    </>
+  ) : (
+    <Root onOpenChange={onOpenChange}>
+      {trigger}
+      <Portal>
+        <Content asChild>
+          <FloatingMenu title={title}>{children}</FloatingMenu>
+        </Content>
+      </Portal>
+    </Root>
+  );
+};
diff --git a/src/components/Menu/ToggleMenuItem.tsx b/src/components/Menu/ToggleMenuItem.tsx
index 71789366..3e5e4ff2 100644
--- a/src/components/Menu/ToggleMenuItem.tsx
+++ b/src/components/Menu/ToggleMenuItem.tsx
@@ -23,7 +23,7 @@ type Props = Pick<
   ComponentProps<typeof MenuItem>,
   "className" | "Icon" | "label" | "onSelect"
 > &
-  Omit<ComponentProps<typeof ToggleInput>, "id" | "children">;
+  Omit<ComponentProps<typeof ToggleInput>, "id" | "children" | "onSelect">;
 
 /**
  * A menu item with a toggle control. Clicking anywhere on the surface will
diff --git a/src/index.ts b/src/index.ts
index 25be450b..1a3b1aca 100644
--- a/src/index.ts
+++ b/src/index.ts
@@ -25,6 +25,7 @@ export { Badge } from "./components/Badge/Badge";
 export { Button, IconButton } from "./components/Button";
 export { Body } from "./components/Typography/Body";
 export { Text } from "./components/Typography/Text";
+export { ContextMenu } from "./components/Menu/ContextMenu";
 export { Glass } from "./components/Glass/Glass";
 export {
   Heading,
diff --git a/vite.config.ts b/vite.config.ts
index 4653a7c9..61830aba 100644
--- a/vite.config.ts
+++ b/vite.config.ts
@@ -34,6 +34,7 @@ export default defineConfig({
         "classnames",
         "graphemer",
         "vaul",
+        "@radix-ui/react-context-menu",
         "@radix-ui/react-dropdown-menu",
         "@radix-ui/react-form",
         "@radix-ui/react-tooltip",
diff --git a/yarn.lock b/yarn.lock
index 87e90e9d..3e3231c4 100644
--- a/yarn.lock
+++ b/yarn.lock
@@ -1747,6 +1747,19 @@
   dependencies:
     "@babel/runtime" "^7.13.10"
 
+"@radix-ui/react-context-menu@^2.1.5":
+  version "2.1.5"
+  resolved "https://registry.yarnpkg.com/@radix-ui/react-context-menu/-/react-context-menu-2.1.5.tgz#1bdbd72761439f9166f75dc4598f276265785c83"
+  integrity sha512-R5XaDj06Xul1KGb+WP8qiOh7tKJNz2durpLBXAGZjSVtctcRFCuEvy2gtMwRJGePwQQE5nV77gs4FwRi8T+r2g==
+  dependencies:
+    "@babel/runtime" "^7.13.10"
+    "@radix-ui/primitive" "1.0.1"
+    "@radix-ui/react-context" "1.0.1"
+    "@radix-ui/react-menu" "2.0.6"
+    "@radix-ui/react-primitive" "1.0.3"
+    "@radix-ui/react-use-callback-ref" "1.0.1"
+    "@radix-ui/react-use-controllable-state" "1.0.1"
+
 "@radix-ui/react-context@1.0.1":
   version "1.0.1"
   resolved "https://registry.yarnpkg.com/@radix-ui/react-context/-/react-context-1.0.1.tgz#fe46e67c96b240de59187dcb7a1a50ce3e2ec00c"