diff --git a/src/humanize/time.py b/src/humanize/time.py
index 4a07d52..5d68c54 100644
--- a/src/humanize/time.py
+++ b/src/humanize/time.py
@@ -75,6 +75,7 @@ def _date_and_delta(
     If that's not possible, return `(None, value)`.
     """
     import datetime as dt
+    import math
 
     if not now:
         now = _now()
@@ -85,11 +86,19 @@ def _date_and_delta(
         date = now - value
         delta = value
     else:
+        # Reject non-finite floats (inf, -inf, NaN) before they reach
+        # round() or dt.timedelta(): `round(float("inf"))` raises
+        # OverflowError, `dt.timedelta(seconds=float("inf"))` raises
+        # OverflowError, and `round(float("nan"))` raises ValueError. The
+        # downstream callers already treat `(None, value)` as "return
+        # str(value) unchanged", which matches the docstring contract.
+        if isinstance(value, float) and not math.isfinite(value):
+            return None, value
         try:
             value = value if precise else round(value)
             delta = dt.timedelta(seconds=value)
             date = now - delta
-        except (ValueError, TypeError):
+        except (ValueError, TypeError, OverflowError):
             return None, value
     return date, _abs_timedelta(delta)
 
@@ -275,6 +284,8 @@ def naturaltime(
 
     Returns:
         str: A natural representation of the input in a resolution that makes sense.
+        Non-finite floats (`inf`, `-inf`, `nan`) and unparseable inputs are
+        returned as their `str()` representation unchanged.
     """
     import datetime as dt
 
diff --git a/tests/test_time.py b/tests/test_time.py
index 7699770..eb3549e 100644
--- a/tests/test_time.py
+++ b/tests/test_time.py
@@ -852,3 +852,61 @@ def test_time_unit() -> None:
 )
 def test_rounding_by_fmt(fmt: str, value: float, expected: float) -> None:
     assert time._rounding_by_fmt(fmt, value) == pytest.approx(expected)
+
+
+@pytest.mark.parametrize(
+    "value, expected",
+    [
+        (float("inf"), "inf"),
+        (float("-inf"), "-inf"),
+        (float("nan"), "nan"),
+    ],
+)
+def test_naturaltime_non_finite_returns_str(value: float, expected: str) -> None:
+    """naturaltime(float) must not crash on non-finite floats.
+
+    Pre-fix, naturaltime(float('inf')) and naturaltime(float('-inf')) raised
+    OverflowError: cannot convert float infinity to integer from inside
+    _date_and_delta's round() call. NaN was already handled because
+    round(float('nan')) raises ValueError, which the except clause caught.
+    """
+    assert humanize.naturaltime(value) == expected
+    assert humanize.naturaltime(value, future=True) == expected
+
+
+@pytest.mark.parametrize(
+    "value, expected",
+    [
+        (float("inf"), "inf"),
+        (float("-inf"), "-inf"),
+        (float("nan"), "nan"),
+    ],
+)
+def test_precisedelta_non_finite_returns_str(value: float, expected: str) -> None:
+    """precisedelta(float) must not crash on non-finite floats.
+
+    Pre-fix, precisedelta(float('inf')) and precisedelta(float('-inf'))
+    raised OverflowError: cannot convert float infinity to integer from
+    _date_and_delta(value, precise=True). NaN was already handled.
+    """
+    assert humanize.precisedelta(value) == expected
+    assert humanize.precisedelta(value, minimum_unit="microseconds") == expected
+
+
+@pytest.mark.parametrize(
+    "value",
+    [float("inf"), float("-inf"), float("nan")],
+)
+def test_date_and_delta_non_finite_returns_none_tuple(value: float) -> None:
+    """_date_and_delta must return (None, value) for non-finite floats.
+
+    This is the contract that the public-facing naturaltime/precisedelta
+    functions rely on to fall through to return str(value).
+    """
+    date, delta = time._date_and_delta(value)
+    assert date is None
+    assert delta is value
+    # precise=True exercises the precisedelta code path
+    date_p, delta_p = time._date_and_delta(value, precise=True)
+    assert date_p is None
+    assert delta_p is value