diff --git a/starlette/applications.py b/starlette/applications.py
index 4542f73c..013364be 100644
--- a/starlette/applications.py
+++ b/starlette/applications.py
@@ -11,6 +11,8 @@ from starlette.responses import Response
 from starlette.routing import BaseRoute, Router
 from starlette.types import ASGIApp, Lifespan, Receive, Scope, Send
 
+AppType = typing.TypeVar("AppType", bound="Starlette")
+
 
 class Starlette:
     """
@@ -43,7 +45,7 @@ class Starlette:
     """
 
     def __init__(
-        self,
+        self: "AppType",
         debug: bool = False,
         routes: typing.Optional[typing.Sequence[BaseRoute]] = None,
         middleware: typing.Optional[typing.Sequence[Middleware]] = None,
@@ -58,7 +60,7 @@ class Starlette:
         ] = None,
         on_startup: typing.Optional[typing.Sequence[typing.Callable]] = None,
         on_shutdown: typing.Optional[typing.Sequence[typing.Callable]] = None,
-        lifespan: typing.Optional[Lifespan] = None,
+        lifespan: typing.Optional[Lifespan["AppType"]] = None,
     ) -> None:
         # The lifespan context function is a newer style that replaces
         # on_startup / on_shutdown handlers. Use one or the other, not both.
diff --git a/starlette/routing.py b/starlette/routing.py
index 024c771b..52cf174e 100644
--- a/starlette/routing.py
+++ b/starlette/routing.py
@@ -580,7 +580,9 @@ class Router:
         default: typing.Optional[ASGIApp] = None,
         on_startup: typing.Optional[typing.Sequence[typing.Callable]] = None,
         on_shutdown: typing.Optional[typing.Sequence[typing.Callable]] = None,
-        lifespan: typing.Optional[Lifespan] = None,
+        # the generic to Lifespan[AppType] is the type of the top level application
+        # which the router cannot know statically, so we use typing.Any
+        lifespan: typing.Optional[Lifespan[typing.Any]] = None,
     ) -> None:
         self.routes = [] if routes is None else list(routes)
         self.redirect_slashes = redirect_slashes
diff --git a/starlette/types.py b/starlette/types.py
index 05f5446e..713d18a8 100644
--- a/starlette/types.py
+++ b/starlette/types.py
@@ -1,7 +1,6 @@
 import typing
 
-if typing.TYPE_CHECKING:
-    from starlette.applications import Starlette
+AppType = typing.TypeVar("AppType")
 
 Scope = typing.MutableMapping[str, typing.Any]
 Message = typing.MutableMapping[str, typing.Any]
@@ -11,8 +10,8 @@ Send = typing.Callable[[Message], typing.Awaitable[None]]
 
 ASGIApp = typing.Callable[[Scope, Receive, Send], typing.Awaitable[None]]
 
-StatelessLifespan = typing.Callable[["Starlette"], typing.AsyncContextManager[None]]
+StatelessLifespan = typing.Callable[[AppType], typing.AsyncContextManager[None]]
 StatefulLifespan = typing.Callable[
-    ["Starlette"], typing.AsyncContextManager[typing.Mapping[str, typing.Any]]
+    [AppType], typing.AsyncContextManager[typing.Mapping[str, typing.Any]]
 ]
-Lifespan = typing.Union[StatelessLifespan, StatefulLifespan]
+Lifespan = typing.Union[StatelessLifespan[AppType], StatefulLifespan[AppType]]
diff --git a/tests/test_applications.py b/tests/test_applications.py
index ef3f7900..e30ec929 100644
--- a/tests/test_applications.py
+++ b/tests/test_applications.py
@@ -1,6 +1,6 @@
 import os
 from contextlib import asynccontextmanager
-from typing import Any, Callable
+from typing import Any, AsyncIterator, Callable
 
 import anyio
 import httpx
@@ -534,3 +534,17 @@ def test_middleware_stack_init(test_client_factory: Callable[[ASGIApp], httpx.Cl
     test_client_factory(app).get("/foo")
 
     assert SimpleInitializableMiddleware.counter == 2
+
+
+def test_lifespan_app_subclass():
+    # This test exists to make sure that subclasses of Starlette
+    # (like FastAPI) are compatible with the types hints for Lifespan
+
+    class App(Starlette):
+        pass
+
+    @asynccontextmanager
+    async def lifespan(app: App) -> AsyncIterator[None]:  # pragma: no cover
+        yield
+
+    App(lifespan=lifespan)