The moduli space of representations of the fundamental group of a compact surface in a Lie group has a tremendously rich geometry relating to symplectic geometry, hyperkähler geometry, integrable systems, Teichmüller theory, etc. This space can be interpreted in purely differential-geometric terms as a moduli space of flat connections. Moreover, by choosing a complex structure on the surface, it can be identified with a moduli space of holomorphic objects known as Higgs bundles over the Riemann surface. After explaining these correspondences, we will illustrate the power of the holomorphic point of view to study the topology of the moduli spaces.