  In this paper we are interested in the asymptotic enumeration of bipartite
Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian
group and let $\iota$ be the automorphism of $A$ defined by $a^\iota=a^{-1}$,
for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism
group as small as possible if $\Aut(\Cay(A,S)) = \langle A,\iota\rangle$. In
this paper, we show that, except for two infinite families, almost all
bipartite Cayley graphs on abelian groups have automorphism group as small as
possible. We also investigate the analogous question for bipartite Cayley
digraphs.
