Traditional black holes, as predicted by Albert Einstein's theory of General Relativity, contain what are known as singularities, i.e., points where the laws of physics break down. Identifying how singularities are resolved in the context of quantum gravity is one of the fundamental problems in theoretical physics.
Depends what you mean by “our current theories”. In classical General Relatively the answer is pretty conclusively no but many people think that a quantum theory of gravity should be able to remove the singularities. In fact, this article is about an attempt to do just that with a fairly natural extension to GR (albeit one that is only mathematically tractable in 5 or more dimensions) and seems to have succeeded for the static spherically symmetric case at least.