Let X be a compact Riemannian surface and let G be a semisimple Lie group. The G-character variety is the quocient R/G of the space of reductive representations of the fundamental group of X in G by the action of conjugation of G. This is a space with very rich geometry and topology, both reflecting the geometry and topology of the surface X as well as of the group G. We shall see how to use holomorphic G-Higgs bundles over X in order to determine the simplest topological invariant of R/G, namely the number of its connected components.