Given a sequence of finite volume locally symmetric spaces $\Gamma\backslash X$ we can consider its Benjamini-Schramm limit. This is a probability measure on the space of pointed locally symmetric spaces that captures the geometry aroud a typical point. It is expected that for pairwise non isometric congruence arithmetic orbifolds the only possible B-S limit is the Dirac delta at the universal cover $X$. I proved it in the case of hyperbolic 2 or 3 manifolds. The results in this direction also allow for some control on the complexity of the homotopy type of arithmetic locally symmetric spaces. In my talk I will review the 2 and 3 dimensional hyperbolic cases and discuss recent progress in the case of general locally symmetric spaces (based on j.w. in progress with Sebastian Hurtado and Jean Raimbault)