diff --git a/src/mcp/server/lowlevel/server.py b/src/mcp/server/lowlevel/server.py
index 643e1a2..27ca276 100644
--- a/src/mcp/server/lowlevel/server.py
+++ b/src/mcp/server/lowlevel/server.py
@@ -68,9 +68,10 @@ import contextvars
 import logging
 import warnings
 from collections.abc import Awaitable, Callable
-from contextlib import AbstractAsyncContextManager, asynccontextmanager
+from contextlib import AbstractAsyncContextManager, AsyncExitStack, asynccontextmanager
 from typing import Any, AsyncIterator, Generic, Sequence, TypeVar
 
+import anyio
 from anyio.streams.memory import MemoryObjectReceiveStream, MemoryObjectSendStream
 from pydantic import AnyUrl
 
@@ -458,6 +459,30 @@ class Server(Generic[LifespanResultT]):
 
         return decorator
 
+    async def _handle_message(
+        self,
+        message: RequestResponder[types.ClientRequest, types.ServerResult]
+        | types.ClientNotification
+        | Exception,
+        session: ServerSession,
+        lifespan_context: LifespanResultT,
+        raise_exceptions: bool = False,
+    ):
+        with warnings.catch_warnings(record=True) as w:
+            match message:
+                case (
+                    RequestResponder(request=types.ClientRequest(root=req)) as responder
+                ):
+                    with responder:
+                        await self._handle_request(
+                            message, req, session, lifespan_context, raise_exceptions
+                        )
+                case types.ClientNotification(root=notify):
+                    await self._handle_notification(notify)
+
+            for warning in w:
+                logger.info(f"Warning: {warning.category.__name__}: {warning.message}")
+
     async def run(
         self,
         read_stream: MemoryObjectReceiveStream[types.JSONRPCMessage | Exception],
@@ -469,41 +494,23 @@ class Server(Generic[LifespanResultT]):
         # in-process servers.
         raise_exceptions: bool = False,
     ):
-        with warnings.catch_warnings(record=True) as w:
-            from contextlib import AsyncExitStack
-
-            async with AsyncExitStack() as stack:
-                lifespan_context = await stack.enter_async_context(self.lifespan(self))
-                session = await stack.enter_async_context(
-                    ServerSession(read_stream, write_stream, initialization_options)
-                )
+        async with AsyncExitStack() as stack:
+            lifespan_context = await stack.enter_async_context(self.lifespan(self))
+            session = await stack.enter_async_context(
+                ServerSession(read_stream, write_stream, initialization_options)
+            )
 
+            async with anyio.create_task_group() as tg:
                 async for message in session.incoming_messages:
                     logger.debug(f"Received message: {message}")
 
-                    match message:
-                        case (
-                            RequestResponder(
-                                request=types.ClientRequest(root=req)
-                            ) as responder
-                        ):
-                            with responder:
-                                await self._handle_request(
-                                    message,
-                                    req,
-                                    session,
-                                    lifespan_context,
-                                    raise_exceptions,
-                                )
-                        case types.ClientNotification(root=notify):
-                            await self._handle_notification(notify)
-
-                    for warning in w:
-                        logger.info(
-                            "Warning: %s: %s",
-                            warning.category.__name__,
-                            warning.message,
-                        )
+                    tg.start_soon(
+                        self._handle_message,
+                        message,
+                        session,
+                        lifespan_context,
+                        raise_exceptions,
+                    )
 
     async def _handle_request(
         self,