  A Beauville surface is a rigid surface of general type arising as a quotient
of a product of curves $C_{1}$, $C_{2}$ of genera $g_{1},g_{2}\ge 2$ by the
free action of a finite group $G$. In this paper we study those Beauville
surfaces for which $G$ is abelian (so that $G\cong \mathbb{Z}_{n}^{2}$ with
$\gcd(n,6)=1$ by a result of Catanese). For each such $n$ we are able to
describe all such surfaces, give a formula for the number of their isomorphism
classes and identify their possible automorphism groups. This explicit
description also allows us to observe that such surfaces are all defined over
$\mathbb{Q}$.
