It has been well-known since Riemann that there are continuous families of Riemann surfaces of genus g ("the moduli space of curves"). Around 1960 Selberg made the remarkable discovery that in higher dimensions (SL(n,R) instead of SL(2,R)) this is no longer true. Works of Calabi-Vesentini, and especially Weil extended this showing that indeed SL(2,R) is the only exceptional case for local rigidity.
In the early 1970s the stronger rigidity theorems of Mostow and of Margulis were proved. In particular, Mostow found the startling result that in dim>2, the fundamental group actually determines the hyperbolic manifold.
I will outline some of the ideas behind these landmark achievements, as well as describe recent results of Gelander, Margulis and myself, concerning superrigidity for lattices of certain general type acting on uniformly convex spaces. The proof of our results is based on a study of generalized harmonic maps.