Maria Beatrice Pozzetti Ruprecht-Karls-Universität Heidelberg Mathematics An important application of bounded cohomology is the theory of maximal representations: a class of homomorphisms of fundamental groups of Kaehler manifolds (most notably fundamental groups of surfaces and finite volume ball quotients) in Hermitian Lie groups (as Sp(2n,R) or SU(p,q)). I will discuss recent rigidity results for maximal representations of fundamental groups of ball quotients on infinite dimensional symmetric spaces (joint with Duchesne and Lecureux) as well as striking geometric properties of the more flexible maximal representations of fundamental groups of surfaces (joint with Burger).