diff --git a/HISTORY.rst b/HISTORY.rst
index aed47df..ffad389 100644
--- a/HISTORY.rst
+++ b/HISTORY.rst
@@ -4,6 +4,7 @@ History
 
 X.Y.Z (YYYY-MM-DD)
 ------------------
+* Add a weakref.finalize method to close HDUList objects on FitsProxy instances (:pr:`12`)
 * Depend on fsspec (:pr:`11`)
 * Improve dask array name determinism (:pr:`10`)
 * Change license from GPL3 to BSD3 (:pr:`9``)
diff --git a/xarrayfits/fits_proxy.py b/xarrayfits/fits_proxy.py
index 696da74..298db55 100644
--- a/xarrayfits/fits_proxy.py
+++ b/xarrayfits/fits_proxy.py
@@ -54,8 +54,9 @@ def hdu_list(self):
                 try:
                     return self._hdul
                 except AttributeError:
-                    self._hdul = fits.open(self._filename, **self._kwargs)
-                    return self._hdul
+                    self._hdul = hdul = fits.open(self._filename, **self._kwargs)
+                    weakref.finalize(self, hdul.close)
+                    return hdul
 
     def __hash__(self):
         return hash((self._filename, tuple(set(self._kwargs.items()))))