Regular black holes from semi-classical down to Planckian size

In this paper we review various models of curvature singularity free black holes. In the first part of the review we describe semi-classical solutions of the Einstein equations which, however, contains a "quantum" input through the matter source. We start by reviewing the early model by Bardeen where the metric is regularized by-hand through a short-distance cut-off, which is justified in terms of non-linear electro-dynamical effects. This a toy-model model useful to point-out the common features shared by all regular semi-classical black holes. Then, we solve Einstein equations with a Gaussian source encoding the quantum spread of an elementary particle. We identify, the a priori arbitrary, Gaussian width with the Compton wavelength of the quantum particle. This Compton-Gauss model leads to the estimate of a terminal density that a gravitationally collapsed object can achieve. We identify this density to be the Planck density, and reformulate the Gaussian model assuming this as its peak density. All these models, are physically reliable as long as the black hole mass is big enough with respect to the Planck mass. In the truly Planckian regime, the semi-classical approximation breaks down. In this case, a fully quantum black hole description is needed. In the last part of this paper, we propose a non-geometrical quantum model of Planckian black hole implementing the Holographic Principle and realizing the "classicalization" scenario recently introduced by Dvali and collaborators. The classical relation between the mass and radius of the black hole emerges only in the classical limit, far away from the Planck scale.