Abstract: The problem of Equilibration and Thermalization in a closed quantum system is at the forefront of both theoretical and experimental interest. Experiments in cold atom gases make it possible to study coherent quantum dynamics on long time scales. The behavior away from equilibrium and the approach to equilibration and thermalization are thus amenable to being probed experimentally. From the theoretical point of view, in recent years it has been understood that typical quantum systems do locally thermalize because the system serves as a bath for each of its local subsystems, in spite of the fact that as a whole, the system evolves unitarily and cannot thermalize. Local thermalization is understood in terms of typicality of Entanglement. However, such typicality arguments miss the point of what is the essential ingredient for thermalization or lack thereof. In this talk, we show a completely new approach to the study of Entanglement in quantum many-body systems. We show that Entanglement is not just expressed by a single number like that given by entropic measures, but it can be characterized in terms of complexity. Thanks to this approach, we are able to completely classify the approach to equilibrium of different dynamical phases, like ETH, integrable systems, and Many-Body localization.