  A Beauville surface is a complex algebraic surface that can be presented as a
quotient of a product 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". They conjectured that all
finite simple groups, except A5, admit such a structure. This conjecture has
recently been proved by Guralnick-Malle and Fairbairn-Magaard-Parker.
  In this survey we demonstrate another approach towards the proof of this
conjecture, based on probabilistic group theoretical methods, by describing the
following three works. The first is the work of Garion, Larsen and Lubotzky,
showing that the above conjecture holds for almost all finite simple groups of
Lie type. The second is the work of Garion and Penegini on Beauville structures
of alternating groups, based on results of Liebeck and Shalev, and the third is
the case of the group PSL(2,p^e), in which we give bounds on the probability of
generating a Beauville structure. We also discuss other related problems
regarding finite simple quotients of hyperbolic triangle groups and present
some open questions and conjectures.
