Given a finitely generated group F and a complex reductive Lie group G, the G-character variety of F, denoted $X_F G=Hom(F,G)//G$, is typically a singular algebraic variety with interesting geometric and topological properties and appears in many contexts within Mathematical-Physics.
For an important class of such varieties, namely when F is the fundamental group of a Kähler manifold M, $X_F G$ is homeomorphic to a space of G-Higgs bundles over M.
In this seminar, we survey many results on the topology of character varieties concentrating in some simple groups F, such as free or free abelian groups. We also present explicit computations of the E-polynomial (a polynomial generalization of the Euler-Poincaré characteristic) of some character varieties, which also encode and relate to arithmetic properties of these varieties and with spaces of quiver representations, finishing with some conjectures and open problems.