Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let
U
U
be a smooth connected complex algebraic variety and let
f
:
U
→
C
∗
f\colon U\to \mathbb {C}^*
be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of
C
∗
\mathbb {C}^*
by
f
f
gives rise to an infinite cyclic cover
U
f
U^f
of
U
U
. The action of the deck group
Z
\mathbb {Z}
on
U
f
U^f
induces a
Q
[
t
±
1
]
\mathbb {Q}[t^{\pm 1}]
-module structure on
H
∗
(
U
f
;
Q
)
H_*(U^f;\mathbb {Q})
. We show that the torsion parts
A
∗
(
U
f
;
Q
)
A_*(U^f;\mathbb {Q})
of the Alexander modules
H
∗
(
U
f
;
Q
)
H_*(U^f;\mathbb {Q})
carry canonical
Q
\mathbb {Q}
-mixed Hodge structures. We also prove that the covering map
U
f
→
U
U^f \to U
induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of
A
∗
(
U
f
;
Q
)
A_*(U^f;\mathbb {Q})
, as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when
f
:
U
→
C
∗
f\colon U\to \mathbb {C}^*
is proper, we prove the semisimplicity and purity of
A
∗
(
U
f
;
Q
)
A_*(U^f;\mathbb {Q})
, and we compare our mixed Hodge structure on
A
∗
(
U
f
;
Q
)
A_*(U^f;\mathbb {Q})
with the limit mixed Hodge structure on the generic fiber of
f
f
.