The formation of dark matter structures in our Universe can be explained by the standard cosmological model, but the populations of galaxies observed in the distant and nearby Universe pose major challenges to our understanding of galaxy formation. There is increasing recognition that the visible, baryonic part of galaxies does not passively follow the hierarchical build-up of dark halos. A large part of the baryons can be accreted from cold gas flows along the cosmic web. The evolution of galaxies could then be mostly driven by their internal evolution, in addition to interactions and mergers. Many scall-scale processes with major effects on galaxy evolution have been unveiled. They have, however, been studied mostly one by one, ignoring the large-scale cosmological environment. Conversely, cosmological models do not resolve the small-scale internal processes properly yet. This dramatically limits our understanding of galaxy formation. The project is to develop an multi-scale understanding of galaxy formation. We will build comprehensive numerical models of the small-scale gas physics and star formation processes in, and incorporate them in large-scale cosmological simulations. Taking benefit from the best forthcoming computing facilities, this will develop a new understanding of the role of internal physics and external processes in structuring galaxies. Theoretical predictions will be confronted to observations, preparing and using the next generation of instruments along the whole duration of the project. Owing to a uniquely comprehensive approach including physical processes at different scales and an original combination of theory, simulation and observation, a new understanding of the evolution of the baryons through cosmic times can emerge from the project.