diff --git a/pooch/processors.py b/pooch/processors.py
index 4f775367..28034b44 100644
--- a/pooch/processors.py
+++ b/pooch/processors.py
@@ -10,9 +10,6 @@
 """
 
 import abc
-import bz2
-import gzip
-import lzma
 import os
 import shutil
 import sys
@@ -20,6 +17,26 @@
 from tarfile import TarFile
 from zipfile import ZipFile
 
+# bz2, gzip and lzma are optional features of the Python standard library: a
+# Python interpreter can be built without them (e.g. when the bzip2/lzma/zlib
+# development headers are missing). Importing them at module level would make
+# *any* use of pooch fail on such interpreters, even when no decompression is
+# needed. Guard the imports so the modules are only required when the matching
+# Decompress method is actually used (see GH #468). The ``type: ignore`` marks
+# the ``None`` fallback as intentional for the type checker.
+try:
+    import bz2
+except ImportError:
+    bz2 = None  # type: ignore[assignment]
+try:
+    import gzip
+except ImportError:
+    gzip = None  # type: ignore[assignment]
+try:
+    import lzma
+except ImportError:
+    lzma = None  # type: ignore[assignment]
+
 from .utils import get_logger
 
 
@@ -342,6 +359,14 @@ class Decompress:
         "bzip2": bz2,
     }
     extensions: typing.ClassVar = {".xz": "lzma", ".gz": "gzip", ".bz2": "bzip2"}
+    # Name of the standard-library module backing each method, used to give a
+    # clear error when that (optional) module isn't available (see GH #468).
+    module_names: typing.ClassVar = {
+        "lzma": "lzma",
+        "xz": "lzma",
+        "gzip": "gzip",
+        "bzip2": "bz2",
+    }
 
     def __init__(self, method="auto", name=None):
         self.method = method
@@ -417,5 +442,18 @@ def _compression_module(self, fname):
                 if ext in {".zip", ".tar"}:
                     message = " ".join([message, error_archives])
                 raise ValueError(message)
-            return self.modules[self.extensions[ext]]
-        return self.modules[self.method]
+            method = self.extensions[ext]
+        else:
+            method = self.method
+        module = self.modules[method]
+        if module is None:
+            module_name = self.module_names[method]
+            message = (
+                f"Could not decompress '{fname}' because the '{module_name}' "
+                "module is not available in this Python installation. This "
+                f"usually means Python was built without '{module_name}' "
+                "support. Rebuild or reinstall Python with the required "
+                "support to use this compression method."
+            )
+            raise ValueError(message)
+        return module
diff --git a/pooch/tests/test_processors.py b/pooch/tests/test_processors.py
index 0a37593f..f5a58cec 100644
--- a/pooch/tests/test_processors.py
+++ b/pooch/tests/test_processors.py
@@ -8,13 +8,15 @@
 Test the processor hooks
 """
 
+import builtins
+import importlib
 import re
 from pathlib import Path
 from tempfile import TemporaryDirectory
 
 import pytest
 
-from .. import Pooch
+from .. import Pooch, processors
 from ..processors import Decompress, Untar, Unzip
 from .utils import capture_log, check_tiny_data, pooch_test_registry, pooch_test_url
 
@@ -95,6 +97,50 @@ def test_decompress_fails():
         assert "pooch.Unzip/Untar" in exception.value.args[0]
 
 
+@pytest.mark.parametrize(
+    ("method", "fname", "module_name"),
+    [
+        ("lzma", "data.xz", "lzma"),
+        ("xz", "data.xz", "lzma"),
+        ("gzip", "data.gz", "gzip"),
+        ("auto", "data.gz", "gzip"),
+        ("bzip2", "data.bz2", "bz2"),
+        ("auto", "data.bz2", "bz2"),
+    ],
+)
+def test_decompress_unavailable_module(monkeypatch, method, fname, module_name):
+    "A clear error should be raised when the compression module is unavailable"
+    # Simulate a Python built without the optional module (see GH #468)
+    monkeypatch.setitem(Decompress.modules, "lzma", None)
+    monkeypatch.setitem(Decompress.modules, "xz", None)
+    monkeypatch.setitem(Decompress.modules, "gzip", None)
+    monkeypatch.setitem(Decompress.modules, "bzip2", None)
+    processor = Decompress(method=method)
+    with pytest.raises(ValueError, match=re.escape(f"'{module_name}' module")):
+        processor._compression_module(fname)
+
+
+def test_processors_import_without_optional_modules(monkeypatch):
+    "Importing pooch must not fail when bz2/lzma are missing (see GH #468)"
+    real_import = builtins.__import__
+
+    def fake_import(name, *args, **kwargs):
+        if name in ("lzma", "_lzma", "gzip", "_gzip", "bz2", "_bz2"):
+            message = f"No module named '{name}'"
+            raise ModuleNotFoundError(message)
+        return real_import(name, *args, **kwargs)
+
+    monkeypatch.setattr(builtins, "__import__", fake_import)
+    try:
+        reloaded_processors = importlib.reload(processors)
+        assert reloaded_processors.lzma is None
+        assert reloaded_processors.gzip is None
+        assert reloaded_processors.bz2 is None
+    finally:
+        monkeypatch.setattr(builtins, "__import__", real_import)
+        importlib.reload(processors)
+
+
 @pytest.mark.network
 @pytest.mark.parametrize(
     "target_path", [None, "some_custom_path"], ids=["default_path", "custom_path"]