The isometry group of a closed hyperbolic n-manifold is finite. We prove that for every n>1 and every finite group G there is an n-dimensional closed hyperbolic manifold whose isometry group is G. This resolves a longstanding problem whose low dimensional cases n=2 and n=3 were proved by Greenberg ('74) and Kojima ('88) resp. The proof is nonconstructive; it uses a 'probabilistic method', i.e. counting results from the theory of 'subgroup growth'. The talk won't assume any prior knowledge on the subject. Joint work with M. Beliopetsky.