  A Beauville surface is a rigid complex surface of the form (C1 x C2)/G, where
C1 and C2 are non-singular, projective, higher genus curves, and G is a finite
group acting freely on the product. Bauer, Catanese, and Grunewald conjectured
that every finite simple group G, with the exception of A5, gives rise to such
a surface. We prove that this is so for almost all finite simple groups (i.e.,
with at most finitely many exceptions). The proof makes use of the structure
theory of finite simple groups, probability theory, and character estimates.
