"Discovering the nature of dark matter is one of the most significant challenges for modern astrophysics. The favourite theoretical model, the Cold Dark Matter (CDM) paradigm, assumes that dark matter is made of collisionless particles acting on galactic scales purely through gravitational interactions.Despite that the CDM model is remarkably successful in reproducing the large-scale structure of the Universe, it seems to be in tension with observations of low-mass galaxies. The internal dynamics of these galaxies is governed by the dominant dark matter haloes they live in, and would be expected to show the presence of dark matter density cusps, an imprint of the collisionless nature of CDM, instead, they seem to be consistent with the presence of large central dark matter cores.Although the solution to this challenge might lie in our still incomplete understanding of the physics of galaxy formation, it is equally likely that these observations contradict the fundamental hypotheses of the CDM model, and might give us a clue about the underlying dark matter nature. This proposal aims at studying the formation and evolution of galaxies in the context of Self-Interacting Dark Matter (SIDM) where dark matter is self-collisional. This objective will be pursued by pioneering cosmological simulations that couple the collisional behaviour of dark matter to the physics of gas cooling, star formation and supernova feedback to simulate the formation of realistic low-mass galaxies.The completion of this proposal will possibly establish a paradigm shift in dark matter physics and open a new horizon in galaxy formation where dark matter microphysics plays a primary role."