Resistive random access memories (ReRAM) form an emerging type of non-volatile memories, based on an electrically driven resistive switching (RS) of an active material. This Feature Article focuses on a broad class of ReRAM where the active material is a Mott insulator or a correlated system. These materials can indeed undergo various insulator-to-metal transitions (IMT) in response to external perturbations such as electronic doping or temperature. These IMT explain most of resistive switching observed in correlated insulators as, for example, the Joule heating induced RS in VO2. The main part of this Feature Article is dedicated to a new mechanism of resistive switching recently unveiled in canonical Mott insulators such as (V1-xCrx)(2)O-3, NiS2-xSex and AM(4)Q(8) (A = Ga, Ge; M = V, Nb, Ta, Mo; Q = S, Se, Te). In these narrow gap Mott insulators, an electronic avalanche breakdown induces a resistive switching, first volatile above a threshold electric field of a few kV/cm and then non-volatile at higher field. The low resistance state is related to the creation of granular conductive filaments, which, in the non-volatile case, can be erased by means of Joule heating. ReRAM devices based on this new type of out of equilibrium Mott insulator-to-metal transition display promising performances.