Thurston's geometrisation program tells us that we can understand all closed 3-manifolds if we can understand those with a finite volume hyperbolic structure. This is not easy, however, and it is not even clear at the outset that there exist more than a few such manifolds. I will decribe number theoretic methods which construct a rich class of finite volume hyperbolic 3-manifolds, and beautiful converse results which show that all such manifolds have useful arithmetic information associated with them.