Let M be a compact manifold. Does the interior of M admit a `nice' complete Riemannian metric. Uniformization says yes in dimension 2, and in dimension 3 an answer comes via Thurston's geometrization program. E.g., if M is a compact 2- or 3- manifold with torus boundary and the fundamental group of M is sufficient complicated (in a very precise sense), then the interior of M admits a complete hyperbolic metric of finite volume. The higher dimensional situation is far more mysterious. For example, let M be a compact n-manifold with torus boundary. I will explain why, for n > 30, none of our tricks for building hyperbolic manifolds could put a complete hyperbolic structure on M.