We know, thanks to the Weil conjectures, that counting points of varieties over finite fields yields purely topological information about them. In this talk I will first describe how we may count the number of points over finite fields on the character varieties parameterizing certain representations of the fundamental group of a Riemann surface into GL_n. The calculation involves an array of techniques from combinatorics to the representation theory of finite groups of Lie type. I will then discuss the geometric implications of this computation and the conjectures it has led to. This is joint work with T. Hausel and E. Letellier