abstract: Moduli spaces of Higgs bundles appear in Simpson's non-abelian Hodge correspondence as an algebraic interpretation of a first cohomology group that admits an action C, an analog of a Hodge structure. These spaces have a very rich geometry, in particular, the underlying space of complex points has several different structures as points of an algebraic variety. The interplay of these different geometries has given rise to many results and open conjectures. In the lectures we will try to explain some of these.