Any closed oriented topological surface of genus greater than or equal to two has a metric of constant negative curvature. The surface with its metric can be written as a quotient of the hyperbolic plane by a so-called Fuchsian subgroup of the isometry group G=PSL(2,R) of the hyperbolic plane. The Fuchsian group is given by a homomorphism of the fundamental group of the surface into PSL(2,R). The subspace of Fuchsian representations (also known as the Teichmüller space of the surface) forms a connected component in the space of all representations of the surface group. I will review this beautiful chapter of mathematics. I will then explain how one can find analogous subspaces for some more general Lie groups G and how this connects to an emerging area of geometry known as Higher Teichmüller theory. Gadgets called Higgs bundles, and their moduli spaces, will play an important role in the story.