We study the asymptotic behavior of the Betti numbers of higher rank locally symmetric manifolds as their volumes tend to infinity, and prove a uniform version of the Lueck Approximation Theorem, which is much stronger than the linear upper bounds proved by Gromov. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows one to derive the convergence of the normalized Betti numbers. Similarly, and more generally, we show that the normalized multiplicity of any unitary representation converges to its Plancherel measure. For congruence covers of a fixed manifold, we also obtain sharper estimates in terms of the rate of convergence. Joint work with M. Abert, N. Bergeron, I. Biringer, N. Nikolov, J. Raimbault and I. Samet.