Given a compact Riemann surface X and a real reductive Lie group G, let  R_G=Hom^{red}(\pi_1 X,G)/G be the space of reductive representations of \pi_1 X in G, the so-called G-character variety of X. This is a space with a very rich topological structure, reflecting the topology of both X and G. We shall introduce G-Higgs bundles over X and explain how to use them for studying the simplest topological invariant of R_G, namely its connected components. The general intrinsic approach for G complex will mentioned and also some examples such as G=Sp(2p,2q) will be considered.

Based on joint work with Oscar García-Prada (ICMAT).