The aim of this proposal is to deepen our knowledge of the role played by the electroweak interaction in systems bound by the strong interaction. Two main physical processes are due to the interplay of these fundamental forces of nature, and are therefore the focus of this proposal: lepton (electron, neutrino) scattering by atomic nuclei, and nuclear beta decays. Advanced theoretical methods will be used to obtain the structure of the strongly bound systems under study, based on a deformed mean field with residual interactions. The fundamental tenets of the electroweak theory will be applied to the study of lepton scattering and binding by nuclei, as well as to simple and double beta decays, including the neutrinoless mode in the latter case. All these processes have deep implications on hot topics in modern physics, ranging from Particle Physics (neutrino nature, standard model tests, nucleon structure) to Nuclear Physics (neutron distribution, deformation, beta decays), including Astrophysics (stellar nucleosynthesis, structure of neutron stars), Cosmology (neutrino masses and dark matter candidates) and Applied Physics (nuclear medicine, environmental and geophysical issues, nuclear reactors decay heat, etc.). Great experimental effort on the human and on the economic sides is currently being made on these fields (CERN, FAIR, GSI, ...) and therefore reliable theoretical guides as the ones proposed here are a must. They match with some of the recommendations included in the Long Range Plan of the Nuclear Physics European Collaboration Committee (European Science Foundation) for the next decade and beyond.A considerable transfer of knowledge concerning advanced theoretical methods and general research and teaching procedures is intended from the outgoing host institution (MIT Laboratory for Nuclear Science, USA) to the return host organization (Complutense University of Madrid, Spain).