diff --git a/.github/workflows/test.yml b/.github/workflows/test.yml
index 74fe883..596e69a 100644
--- a/.github/workflows/test.yml
+++ b/.github/workflows/test.yml
@@ -61,6 +61,7 @@ jobs:
         if: matrix.anyio-version == 'anyio-v4'
         run: pip install --upgrade "anyio>=4.0.0,<5.0"
       - name: Lint
+        if: matrix.anyio-version == 'anyio-v4'
         run: bash scripts/lint.sh
       - run: mkdir coverage
       - name: Test
diff --git a/asyncer/_compat.py b/asyncer/_compat.py
new file mode 100644
index 0000000..ade175e
--- /dev/null
+++ b/asyncer/_compat.py
@@ -0,0 +1,37 @@
+# AnyIO 4.1.0 renamed cancellable to abandon_on_cancel
+import importlib
+import importlib.metadata
+from typing import Callable, TypeVar, Union
+
+import anyio
+import anyio.to_thread
+from anyio import CapacityLimiter
+from typing_extensions import TypeVarTuple, Unpack
+
+ANYIO_VERSION = importlib.metadata.version("anyio")
+
+T_Retval = TypeVar("T_Retval")
+PosArgsT = TypeVarTuple("PosArgsT")
+
+if ANYIO_VERSION >= "4.1.0":
+
+    async def run_sync(
+        func: Callable[[Unpack[PosArgsT]], T_Retval],
+        *args: Unpack[PosArgsT],
+        abandon_on_cancel: bool = False,
+        limiter: Union[CapacityLimiter, None] = None,
+    ) -> T_Retval:
+        return await anyio.to_thread.run_sync(
+            func, *args, abandon_on_cancel=abandon_on_cancel, limiter=limiter
+        )
+else:
+
+    async def run_sync(
+        func: Callable[[Unpack[PosArgsT]], T_Retval],
+        *args: Unpack[PosArgsT],
+        abandon_on_cancel: bool = False,
+        limiter: Union[CapacityLimiter, None] = None,
+    ) -> T_Retval:
+        return await anyio.to_thread.run_sync(
+            func, *args, cancellable=abandon_on_cancel, limiter=limiter
+        )
diff --git a/asyncer/_main.py b/asyncer/_main.py
index 4eeb5eb..78117e4 100644
--- a/asyncer/_main.py
+++ b/asyncer/_main.py
@@ -12,6 +12,9 @@
     TypeVar,
     Union,
 )
+from warnings import warn
+
+from asyncer._compat import run_sync
 
 if sys.version_info >= (3, 10):
     from typing import ParamSpec
@@ -319,7 +322,8 @@ def wrapper(*args: T_ParamSpec.args, **kwargs: T_ParamSpec.kwargs) -> T_Retval:
 def asyncify(
     function: Callable[T_ParamSpec, T_Retval],
     *,
-    cancellable: bool = False,
+    abandon_on_cancel: bool = False,
+    cancellable: Union[bool, None] = None,
     limiter: Optional[anyio.CapacityLimiter] = None,
 ) -> Callable[T_ParamSpec, Awaitable[T_Retval]]:
     """
@@ -359,14 +363,24 @@ def do_work(arg1, arg2, kwarg1="", kwarg2="") -> str:
     original one, that when called runs the same original function in a thread worker
     and returns the result.
     """
+    if cancellable is not None:
+        abandon_on_cancel = cancellable
+        warn(
+            "The `cancellable=` keyword argument to `asyncer.asyncify()` is "
+            "deprecated since Asyncer 0.0.8, following AnyIO 4.1.0. "
+            "Use `abandon_on_cancel=` instead.",
+            DeprecationWarning,
+            stacklevel=2,
+        )
 
     @functools.wraps(function)
     async def wrapper(
         *args: T_ParamSpec.args, **kwargs: T_ParamSpec.kwargs
     ) -> T_Retval:
         partial_f = functools.partial(function, *args, **kwargs)
-        return await anyio.to_thread.run_sync(
-            partial_f, cancellable=cancellable, limiter=limiter
+
+        return await run_sync(
+            partial_f, abandon_on_cancel=abandon_on_cancel, limiter=limiter
         )
 
     return wrapper
diff --git a/tests/test_param_cancellable.py b/tests/test_param_cancellable.py
new file mode 100644
index 0000000..4c2b0d4
--- /dev/null
+++ b/tests/test_param_cancellable.py
@@ -0,0 +1,37 @@
+import warnings
+
+import anyio
+import pytest
+from asyncer import asyncify
+
+
+def test_cancellable_warns():
+    def do_async_work():
+        return "Hello World!"
+
+    async def main():
+        result = await asyncify(do_async_work, cancellable=True)()
+        return result
+
+    with pytest.warns(DeprecationWarning) as record:
+        result = anyio.run(main)
+    assert isinstance(record[0].message, Warning)
+    assert (
+        "The `cancellable=` keyword argument to `asyncer.asyncify()` is "
+        "deprecated since Asyncer 0.0.8" in record[0].message.args[0]
+    )
+    assert result == "Hello World!"
+
+
+def test_abandon_on_cancel_no():
+    def do_async_work():
+        return "Hello World!"
+
+    async def main():
+        result = await asyncify(do_async_work, abandon_on_cancel=True)()
+        return result
+
+    with warnings.catch_warnings():
+        warnings.simplefilter("error")
+        result = anyio.run(main)
+    assert result == "Hello World!"