Abstract: Fuchsian representations are discrete faithful representations of the fundamental group of a closed surface into PSL(2,R) (the isometry group of the hyperbolic plane), and are in bijection with the Teichmüller space of hyperbolic structures on the surface. Using Higgs bundle techniques, Nigel Hitchin showed that there is a natural generalization of Fuchsian representations into the group PSL(n,R), however the geometry associated these representations remained mysterious until Labourie's work on Anosov representations. The aim of this talk is to motivate (especially to the speaker) why one should work to understand Anosov structures. After setting the scene, most of the time will be spent talking about Fuchsian representations and the geodesic flow on hyperbolic surfaces.