It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher Teichmüller spaces are generalizations of this, which exist in the moduli space of representations of the fundamental group of the surface into certain real simple non-compact Lie groups of higher rank. As for the usual Teichmüller space, these spaces consist entirely  of discrete and faithful representations.  In this talk, I will give a classification of all possible higher rank Teichmüller spaces, and a parametrization of them from a Higgs bundle perspective  (based on joint work with Bradlow, Collier, Gothen and Oliveira).