Abstract: Black holes contain, deep in their interior, theoretical evidence of the failure of general relativity. A series of fundamental results, starting from the 1965 Penrose singularity theorem, proved that physically realistic initial conditions will unavoidably produce a singular black hole spacetime. It is generally expected that a full theory of quantum gravity should remove the singularities that appear in general relativity. However, the lack of proper understanding of the dynamical laws dictating the evolution of spacetime and matter in these extreme situations hinders the extraction of predictions in specific models. I will discuss, in a model-independent manner, the different possibilities that singularity regularization may open, focus on fundamental open issues that need to be addressed to obtain viable nonsingular black hole candidates, and finally discuss some observational signatures.