We consider complete analytic Riemannian manifolds of bounded nonpositive curvature. These include locally symmetric and, in particular, arithmetic manifolds, the study of which is strongly related to the theory of lattices and arithmetic groups. The complexity of such manifolds is controlled by the volume. This phenomenon can be measured in terms of the growth of topological, geometric, algebraic, arithmetic and representation theoretic invariants, such as Betti numbers and torsion, optimal presentations of Γ=π1(M), the minimal size of a triangulation as well as invariant related to the Plancherel measure associated to L_2(G/Γ). Another line of problems concern with the number of manifolds of a certain type and bounded volume. In the talk I will give an overview of the theory.