diff --git a/newsfragments/2493.breaking.rst b/newsfragments/2493.breaking.rst
new file mode 100644
index 0000000000..51c2985d1a
--- /dev/null
+++ b/newsfragments/2493.breaking.rst
@@ -0,0 +1 @@
+Timeout functions now raise `ValueError` if passed `math.nan`. This includes `trio.sleep`, `trio.sleep_until`, `trio.move_on_at`, `trio.move_on_after`, `trio.fail_at` and `trio.fail_after`.
diff --git a/trio/_tests/test_timeouts.py b/trio/_tests/test_timeouts.py
index c817c49588..f55e697d6f 100644
--- a/trio/_tests/test_timeouts.py
+++ b/trio/_tests/test_timeouts.py
@@ -53,9 +53,6 @@ async def sleep_2():
 
     await check_takes_about(sleep_2, TARGET)
 
-    with pytest.raises(ValueError):
-        await sleep(-1)
-
     with assert_checkpoints():
         await sleep(0)
     # This also serves as a test of the trivial move_on_at
@@ -66,10 +63,6 @@ async def sleep_2():
 
 @slow
 async def test_move_on_after():
-    with pytest.raises(ValueError):
-        with move_on_after(-1):
-            pass  # pragma: no cover
-
     async def sleep_3():
         with move_on_after(TARGET):
             await sleep(100)
@@ -99,6 +92,29 @@ async def sleep_5():
     with fail_after(100):
         await sleep(0)
 
-    with pytest.raises(ValueError):
-        with fail_after(-1):
-            pass  # pragma: no cover
+
+async def test_timeouts_raise_value_error():
+    # deadlines are allowed to be negative, but not delays.
+    # neither delays nor deadlines are allowed to be NaN
+
+    nan = float("nan")
+
+    for fun, val in (
+        (sleep, -1),
+        (sleep, nan),
+        (sleep_until, nan),
+    ):
+        with pytest.raises(ValueError):
+            await fun(val)
+
+    for cm, val in (
+        (fail_after, -1),
+        (fail_after, nan),
+        (fail_at, nan),
+        (move_on_after, -1),
+        (move_on_after, nan),
+        (move_on_at, nan),
+    ):
+        with pytest.raises(ValueError):
+            with cm(val):
+                pass  # pragma: no cover
diff --git a/trio/_timeouts.py b/trio/_timeouts.py
index 1f7878f89e..ad31e78404 100644
--- a/trio/_timeouts.py
+++ b/trio/_timeouts.py
@@ -1,3 +1,4 @@
+import math
 from contextlib import contextmanager
 
 import trio
@@ -10,7 +11,12 @@ def move_on_at(deadline):
     Args:
       deadline (float): The deadline.
 
+    Raises:
+      ValueError: if deadline is NaN.
+
     """
+    if math.isnan(deadline):
+        raise ValueError("deadline must not be NaN")
     return trio.CancelScope(deadline=deadline)
 
 
@@ -22,10 +28,9 @@ def move_on_after(seconds):
       seconds (float): The timeout.
 
     Raises:
-      ValueError: if timeout is less than zero.
+      ValueError: if timeout is less than zero or NaN.
 
     """
-
     if seconds < 0:
         raise ValueError("timeout must be non-negative")
     return move_on_at(trio.current_time() + seconds)
@@ -52,6 +57,9 @@ async def sleep_until(deadline):
             the past, in which case this function executes a checkpoint but
             does not block.
 
+    Raises:
+      ValueError: if deadline is NaN.
+
     """
     with move_on_at(deadline):
         await sleep_forever()
@@ -65,7 +73,7 @@ async def sleep(seconds):
             insert a checkpoint without actually blocking.
 
     Raises:
-        ValueError: if *seconds* is negative.
+        ValueError: if *seconds* is negative or NaN.
 
     """
     if seconds < 0:
@@ -96,9 +104,13 @@ def fail_at(deadline):
     :func:`fail_at`, then it's caught and :exc:`TooSlowError` is raised in its
     place.
 
+    Args:
+      deadline (float): The deadline.
+
     Raises:
       TooSlowError: if a :exc:`Cancelled` exception is raised in this scope
         and caught by the context manager.
+      ValueError: if deadline is NaN.
 
     """
 
@@ -119,10 +131,13 @@ def fail_after(seconds):
     it's caught and discarded. When it reaches :func:`fail_after`, then it's
     caught and :exc:`TooSlowError` is raised in its place.
 
+    Args:
+      seconds (float): The timeout.
+
     Raises:
       TooSlowError: if a :exc:`Cancelled` exception is raised in this scope
         and caught by the context manager.
-      ValueError: if *seconds* is less than zero.
+      ValueError: if *seconds* is less than zero or NaN.
 
     """
     if seconds < 0: