Hearts of Darkness: theory and phenomenology of non-singular black holes

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. I will discuss in a model-independent manner the different possibilities that singularity regularization may open. After that, I will focus on consistency issues due to an exponential blue-shift of the perturbation at the inner horizon. This phenomenon goes by the name of mass inflation instability and cast serious doubts on the viability of regular black holes as resolution to the singularity problem. Finally, I will stress the observational implications that stem from this analysis.