Classical Teichmüller theory can be understood as the study of a connected component in the variety parametrising rapresentations from the fundamental group of a topological surface of genus at least 2 in the group PSL_2(R) of isometries of the hyperbolic space. I will discuss joint work with Beyrer-Guichard-Labourie-Wienhard in which we develop a similar theory for some Lie groups G other than PSL_2(R).