In the philosophy of mathematics, some attempts have been made to give it ultimately secure foundations; a notable example is the Hilbert Program. Goedel's Theorems show that it is not quite possible, but there are some partial results. Most such considerations that I know of restrict their attention to the part of mathematics which is axiomatizable in ZFC set theory, possibly enriched by some extra axioms.

Nevertheless, mathematics does not end at ZFC. In particular, we have category theory with proper classes which are not, and cannot be, sets. Are people concerned about the well-foundedness of category theory? Is it possible to show that if ZFC does not produce a contradiction, then neither does category theory? 

The reason I am asking is that I have been wondering how certain I really am of correctness of mathematics. Paradoxes such as Russell's show that we can get basic notions (such as sets) painfully wrong. That we have been doing mathematics in the ZFC framework for some decades now convinces me (on a practical level) that there are no surprises like that left to be discovered in set theory. I am however worried whether we need to keep adding more abstract theories (like category theory) to get the full picture of mathematics, and if there might be paradoxes like Russell's lurking somewhere in there. Could there be?