Abstract: We introduce the simplicial volume of manifolds and explain its applications and relations to other manifold invariants like minimal volume and L^2-Betti numbers. Then we discuss to what extent Gromov's proportionality principle, which compares the simplicial volume of closed Riemannian manifolds with isometric universal covers, holds for open Riemannian manifolds of finite volume. Finally, applications to locally symmetric spaces of finite volume (like e.g. positivity of the minimal volume) are presented. The talk is based on joint work with Clara Loeh.