  We answer a conjecture of Bauer, Catanese and Grunewald showing that all
finite simple groups other than the alternating group of degree 5 admit unmixed
Beauville structures. We also consider an analog of the result for simple
algebraic groups which depends on some upper bounds for character values of
regular semisimple elements in finite groups of Lie type and obtain definitive
results about the variety of triples in semisimple regular classes with product
1. Finally, we prove that any finite simple group contains two conjugacy
classes C,D such that any pair of elements in C x D generates the group.
