An orientable hyperbolic n-manifold is isometric to the quotient of hyper-

bolic n-space H by a discrete torsion free subgroup of the group of

orientation-preserving isometries of H. Among these manifolds, the ones

originating from arithmetically defined groups form a family of special

interest. Due to the underlying connections with number theory and the

theory of automorphic forms, there is a fruitful interaction between

geometric and arithmetic questions, methods and results. We intend to give

an account of recent investigations in this area, in particular, of those

pertaining to hyperbolic 3-manifolds and bounds for their Betti numbers.