Roughly speaking, a Beauville surface is a complex surface which is the quotient of the product of two algebraic curves by a special action of a finite group G, which is called a Beauville group. What makes Beauville surfaces so good to deal with is the fact that they can be described in purely group theoretical terms. It turns out that a group G can be used to define a Beauville surface if and only if it has a certain pair of generating sets known as a Beauville structure.

The question as to which finite groups are Beauville groups has received considerable attention since the early 2000s. This talk will first survey results on Beauville groups in a general context, and will then focus specifically on Beauville p-groups. If the time permits, we will also discuss the number of isomorphism classes of Beauville surfaces from a group-theoretic perspective.