We discuss some open problems and some new results about the topology of arithmetic locally symmetric spaces. Among the new results is a proof of a conjecture of Gelander stating that the complexity of the topology of these manifolds can be bounded just in terms of the volume. The main new tool is an arithmetic refinement of the classical Margulis lemma about discrete subgroups of Lie groups. Based on joint work with Mikolaj Fraczyk and Jean Raimbault. All notions will be explained. Speaker(s): Sebastian Hurtado-Salazar (University of Chicago)