Understanding electronic correlations remains one of the most important challenges in theoretical condensed matter physics. The interaction-induced metal-to-insulator Mott transition plays a major role in many transition metal oxides, f-electron materials and now in quantum optics. Upon doping or application of a strong electric field, strongly correlated Mott metals emerge from the Mott insulators, with fascinating properties. Moreover, the out-of-equilibrium behaviour of these systems is only beginning to be systematically explored experimentally. While these systems strongly challenge the standard concepts and methods of the quantum many-body theory, a new era is progressively unfolding, in which quantitative and detailed comparisons between theory and experiments is becoming possible in strong correlation regimes, even out of equilibrium.The goal of this proposal is to construct, in close contact with experiments and phenomenology, a new generation of theoretical methods and algorithms in order to i) study the new states of matter induced by non-equilibrium phenomena in strongly correlated quantum systems, first in simple models, and then in realistic computations for real materials; ii) elucidate the mystery of high temperature superconductivity. Open source implementations of the methods and algorithms developed during this project will also be provided for a better knowledge diffusion.