Lattices in semi-simple Lie groups of rank at least 2 — e.g. SL(n,Z) for n>2 — form a class of discrete groups known for having remarkable linear rigidity properties. Notably, their finite dimensional representations are determined by those of the ambient Lie group they live in — e.g. SL(n,R) in the case of SL(n,Z). This is Margulis' super-rigidity theorem (1974). Motivated by an ergodic version of this theorem, an ambitious program initiated by Gromov and Zimmer in the 1980s aims to understand "non-linear representations" of such lattices into diffeomorphism groups of closed manifolds, or in other words, the differentiable actions of such lattices on closed manifolds.
I will first discuss the history and geometric origins of this program. I will then focus on rigidity results about actions of lattices which preserve non-unimodular geometric structures, such as conformal or projective structures, and will mention open directions. The proofs build on recent advances on Zimmer's conjectures, especially an invariance principle which provides existence of finite invariant measures in various dynamical contexts.