In this talk I will show how wormholes can resolve black hole singularities without necessarily removing curvature divergences. This is shown by studying geodesic completeness, the behavior of time-like congruences going through the divergent region, and by means of scattering of waves off the wormhole in a particular model. This model is an exact solution of an extension of Einstein's equations coupled to an electric field, i.e., it is not an ad hoc model designed to solve the problem of singularities, even though it does.