diff --git a/docs/versionhistory.rst b/docs/versionhistory.rst
index 5d39e459..db3dcb62 100644
--- a/docs/versionhistory.rst
+++ b/docs/versionhistory.rst
@@ -10,6 +10,18 @@ This library adheres to `Semantic Versioning 2.0 <http://semver.org/>`_.
 - Fixed passing ``total_tokens`` to ``anyio.CapacityLimiter()`` as a keyword argument
   not working on the ``trio`` backend
   (`#515 <https://github.com/agronholm/anyio/issues/515>`_)
+- Fixed ``Process.aclose()`` not performing the minimum level of necessary cleanup when
+  cancelled. Previously:
+
+  - Cancellation of ``Process.aclose()`` could leak an orphan process
+  - Cancellation of ``run_process()`` could very briefly leak an orphan process.
+  - Cancellation of ``Process.aclose()`` or ``run_process()`` on Trio could leave
+    standard streams unclosed
+
+  (PR by Ganden Schaffner)
+- Fixed ``Process.stdin.aclose()``, ``Process.stdout.aclose()``, and
+  ``Process.stderr.aclose()`` not including a checkpoint on asyncio (PR by Ganden
+  Schaffner)
 
 **4.2.0**
 
diff --git a/src/anyio/_backends/_asyncio.py b/src/anyio/_backends/_asyncio.py
index e884f564..2699bf81 100644
--- a/src/anyio/_backends/_asyncio.py
+++ b/src/anyio/_backends/_asyncio.py
@@ -918,6 +918,7 @@ async def receive(self, max_bytes: int = 65536) -> bytes:
 
     async def aclose(self) -> None:
         self._stream.feed_eof()
+        await AsyncIOBackend.checkpoint()
 
 
 @dataclass(eq=False)
@@ -930,6 +931,7 @@ async def send(self, item: bytes) -> None:
 
     async def aclose(self) -> None:
         self._stream.close()
+        await AsyncIOBackend.checkpoint()
 
 
 @dataclass(eq=False)
@@ -940,14 +942,22 @@ class Process(abc.Process):
     _stderr: StreamReaderWrapper | None
 
     async def aclose(self) -> None:
-        if self._stdin:
-            await self._stdin.aclose()
-        if self._stdout:
-            await self._stdout.aclose()
-        if self._stderr:
-            await self._stderr.aclose()
-
-        await self.wait()
+        with CancelScope(shield=True):
+            if self._stdin:
+                await self._stdin.aclose()
+            if self._stdout:
+                await self._stdout.aclose()
+            if self._stderr:
+                await self._stderr.aclose()
+
+        try:
+            await self.wait()
+        except BaseException:
+            self.kill()
+            with CancelScope(shield=True):
+                await self.wait()
+
+            raise
 
     async def wait(self) -> int:
         return await self._process.wait()
diff --git a/src/anyio/_backends/_trio.py b/src/anyio/_backends/_trio.py
index 13b960f2..1a47192e 100644
--- a/src/anyio/_backends/_trio.py
+++ b/src/anyio/_backends/_trio.py
@@ -283,14 +283,21 @@ class Process(abc.Process):
     _stderr: abc.ByteReceiveStream | None
 
     async def aclose(self) -> None:
-        if self._stdin:
-            await self._stdin.aclose()
-        if self._stdout:
-            await self._stdout.aclose()
-        if self._stderr:
-            await self._stderr.aclose()
-
-        await self.wait()
+        with CancelScope(shield=True):
+            if self._stdin:
+                await self._stdin.aclose()
+            if self._stdout:
+                await self._stdout.aclose()
+            if self._stderr:
+                await self._stderr.aclose()
+
+        try:
+            await self.wait()
+        except BaseException:
+            self.kill()
+            with CancelScope(shield=True):
+                await self.wait()
+            raise
 
     async def wait(self) -> int:
         return await self._process.wait()
diff --git a/src/anyio/_core/_subprocesses.py b/src/anyio/_core/_subprocesses.py
index c1610296..5d5d7b76 100644
--- a/src/anyio/_core/_subprocesses.py
+++ b/src/anyio/_core/_subprocesses.py
@@ -65,20 +65,18 @@ async def drain_stream(stream: AsyncIterable[bytes], index: int) -> None:
         start_new_session=start_new_session,
     ) as process:
         stream_contents: list[bytes | None] = [None, None]
-        try:
-            async with create_task_group() as tg:
-                if process.stdout:
-                    tg.start_soon(drain_stream, process.stdout, 0)
-                if process.stderr:
-                    tg.start_soon(drain_stream, process.stderr, 1)
-                if process.stdin and input:
-                    await process.stdin.send(input)
-                    await process.stdin.aclose()
-
-                await process.wait()
-        except BaseException:
-            process.kill()
-            raise
+        async with create_task_group() as tg:
+            if process.stdout:
+                tg.start_soon(drain_stream, process.stdout, 0)
+
+            if process.stderr:
+                tg.start_soon(drain_stream, process.stderr, 1)
+
+            if process.stdin and input:
+                await process.stdin.send(input)
+                await process.stdin.aclose()
+
+            await process.wait()
 
     output, errors = stream_contents
     if check and process.returncode != 0:
diff --git a/tests/test_subprocesses.py b/tests/test_subprocesses.py
index 640384c0..cc54fc5e 100644
--- a/tests/test_subprocesses.py
+++ b/tests/test_subprocesses.py
@@ -8,8 +8,9 @@
 from textwrap import dedent
 
 import pytest
+from _pytest.fixtures import FixtureRequest
 
-from anyio import open_process, run_process
+from anyio import CancelScope, ClosedResourceError, open_process, run_process
 from anyio.streams.buffered import BufferedByteReceiveStream
 
 pytestmark = pytest.mark.anyio
@@ -176,3 +177,61 @@ async def test_run_process_inherit_stdout(capfd: pytest.CaptureFixture[str]) ->
     out, err = capfd.readouterr()
     assert out == "stdout-text" + os.linesep
     assert err == "stderr-text" + os.linesep
+
+
+async def test_process_aexit_cancellation_doesnt_orphan_process() -> None:
+    """
+    Regression test for #669.
+
+    Ensures that open_process.__aexit__() doesn't leave behind an orphan process when
+    cancelled.
+
+    """
+    with CancelScope() as scope:
+        async with await open_process(
+            [sys.executable, "-c", "import time; time.sleep(1)"]
+        ) as process:
+            scope.cancel()
+
+    assert process.returncode is not None
+    assert process.returncode != 0
+
+
+async def test_process_aexit_cancellation_closes_standard_streams(
+    request: FixtureRequest,
+    anyio_backend_name: str,
+) -> None:
+    """
+    Regression test for #669.
+
+    Ensures that open_process.__aexit__() closes standard streams when cancelled. Also
+    ensures that process.std{in.send,{out,err}.receive}() raise ClosedResourceError on a
+    closed stream.
+
+    """
+    if anyio_backend_name == "asyncio":
+        # Avoid pytest.xfail here due to https://github.com/pytest-dev/pytest/issues/9027
+        request.node.add_marker(
+            pytest.mark.xfail(reason="#671 needs to be resolved first")
+        )
+
+    with CancelScope() as scope:
+        async with await open_process(
+            [sys.executable, "-c", "import time; time.sleep(1)"]
+        ) as process:
+            scope.cancel()
+
+    assert process.stdin is not None
+
+    with pytest.raises(ClosedResourceError):
+        await process.stdin.send(b"foo")
+
+    assert process.stdout is not None
+
+    with pytest.raises(ClosedResourceError):
+        await process.stdout.receive(1)
+
+    assert process.stderr is not None
+
+    with pytest.raises(ClosedResourceError):
+        await process.stderr.receive(1)
diff --git a/tests/test_taskgroups.py b/tests/test_taskgroups.py
index f4c87b39..bc4a289d 100644
--- a/tests/test_taskgroups.py
+++ b/tests/test_taskgroups.py
@@ -185,6 +185,9 @@ async def taskfunc(*, task_status: TaskStatus) -> None:
     assert not finished
 
 
+@pytest.mark.xfail(
+    sys.version_info < (3, 9), reason="Requires a way to detect cancellation source"
+)
 @pytest.mark.parametrize("anyio_backend", ["asyncio"])
 async def test_start_native_host_cancelled() -> None:
     started = finished = False
@@ -199,9 +202,6 @@ async def start_another() -> None:
         async with create_task_group() as tg:
             await tg.start(taskfunc)
 
-    if sys.version_info < (3, 9):
-        pytest.xfail("Requires a way to detect cancellation source")
-
     task = asyncio.get_running_loop().create_task(start_another())
     await wait_all_tasks_blocked()
     task.cancel()