Beauville surfaces are certain complex algebraic surfaces which can be described as quotients of products of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups appearing in minimal presentations of Beauville surfaces in terms of the existence of a so-called "Beauville structure", and conjectured that all finite simple groups, except A5, admit such a structure. In the talk I will describe two results. The first is a joint work with Michael Larsen and Alex Lubotzky, showing that the conjecture of Bauer, Catanese and Grunewald holds for almost all finite simple groups of Lie type. The second is a joint work with Matteo Penegini on Beauville structures of alternating groups, based on results of Liebeck and Shalev. The proofs rely on probabilistic group-theoretical methods and character estimates in finite simple groups.