diff --git a/src/humanize/time.py b/src/humanize/time.py
index 4a07d52..37332ee 100644
--- a/src/humanize/time.py
+++ b/src/humanize/time.py
@@ -7,6 +7,7 @@
 
 __lazy_modules__ = {"humanize.i18n", "humanize.number"}
 
+import math
 from enum import Enum
 from functools import total_ordering
 
@@ -91,6 +92,10 @@ def _date_and_delta(
             date = now - delta
         except (ValueError, TypeError):
             return None, value
+        except OverflowError:
+            if not math.isfinite(value):
+                return None, value
+            raise
     return date, _abs_timedelta(delta)
 
 
@@ -153,6 +158,10 @@ def naturaldelta(
             delta = dt.timedelta(seconds=value)
         except (ValueError, TypeError):
             return str(value)
+        except OverflowError:
+            if not math.isfinite(value):
+                return str(value)
+            raise
 
     use_months = months
 
diff --git a/tests/test_time.py b/tests/test_time.py
index 7699770..f1f44d7 100644
--- a/tests/test_time.py
+++ b/tests/test_time.py
@@ -3,6 +3,7 @@
 from __future__ import annotations
 
 import datetime as dt
+import math
 import typing
 
 import pytest
@@ -71,6 +72,8 @@ def test_date_and_delta() -> None:
             assert_equal_datetime(date, result[0])
             assert_equal_timedelta(d, result[1])
     assert time._date_and_delta("NaN") == (None, "NaN")
+    assert time._date_and_delta(float("inf")) == (None, float("inf"))
+    assert time._date_and_delta(float("-inf")) == (None, float("-inf"))
 
 
 # Tests for the public interface of humanize.time
@@ -89,6 +92,12 @@ def test_naturaldelta_nomonths(test_input: dt.timedelta, expected: str) -> None:
     assert humanize.naturaldelta(test_input, months=False) == expected
 
 
+def test_naturaldelta_too_large_value_raises() -> None:
+    """A too-large finite value still raises OverflowError (unlike inf)."""
+    with pytest.raises(OverflowError):
+        humanize.naturaldelta(1e30)
+
+
 @pytest.mark.parametrize(
     "test_input, expected",
     [
@@ -128,13 +137,17 @@ def test_naturaldelta_nomonths(test_input: dt.timedelta, expected: str) -> None:
         (dt.timedelta(days=365), "a year"),
         (dt.timedelta(days=365 * 1_141), "1,141 years"),
         ("NaN", "NaN"),  # Returns non-numbers unchanged.
+        (float("inf"), "inf"),
+        (float("-inf"), "-inf"),
         # largest possible timedelta
         (dt.timedelta(days=999_999_999), "2,739,726 years"),
     ],
 )
 def test_naturaldelta(test_input: float | dt.timedelta, expected: str) -> None:
     assert humanize.naturaldelta(test_input) == expected
-    if not isinstance(test_input, str):
+    if not isinstance(test_input, str) and not (
+        isinstance(test_input, float) and math.isinf(test_input)
+    ):
         assert humanize.naturaldelta(-test_input) == expected
 
 
@@ -179,6 +192,8 @@ def test_naturaldelta(test_input: float | dt.timedelta, expected: str) -> None:
         (NOW - dt.timedelta(days=365 * 2 + 65), "2 years ago"),
         (NOW - dt.timedelta(days=365 + 4), "1 year, 4 days ago"),
         ("NaN", "NaN"),
+        (float("inf"), "inf"),
+        (float("-inf"), "-inf"),
     ],
 )
 def test_naturaltime(
@@ -817,6 +832,8 @@ def test_precisedelta_suppress_units(
 
 def test_precisedelta_bogus_call() -> None:
     assert humanize.precisedelta(None) == "None"
+    assert humanize.precisedelta(float("inf")) == "inf"
+    assert humanize.precisedelta(float("-inf")) == "-inf"
 
     with pytest.raises(
         ValueError,