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 $\Gamma=\pi_1(M)$, the minimal size of a triangulation as well as
invariants related to the Plancherel measure associated to
$L_2(G/\Gamma)$. Other problems concern the number of manifolds of a
certain type and bounded volume. In the talk I will give an overview
of the theory.