Observations of unprecedented quality reveal a Universe that is at tension with the standard, and very successful description of matter and energy in Physics. Around 95% of the substratum of the Universe is of unknown nature, split into an accreting component (dark matter) and a repelling component (dubbed dark energy). There are auspicious prospects that the combination of state-of-the-art experiments, and theoretical advances will provide us with tools to elucidate this fundamental issue. This Action explores the viewpoint that cosmological observations reveal a degree of incongruous with theory not because of mysterious elements, but because of a need to review and extend Einstein Relativity to scales where it has not been properly tested. So this Action “CANTATA” gathers a team of European leading experts in gravitational physics and cosmology around the timely goal of investigating the extension of Einstein’s theory of General Relativity. A program including complementary aspects of theoretical physics, cosmology and astrophysics is put forward which is set to consider, in a coordinated and multidisciplinary way, the build up self-consistent models at the various scales and, in principle, to find out some “crucial feature” capable of confirming or ruling out Extended Theories of Gravity with respect to General Relativity. This Action will enhance already existing collaborations and establish an European network with the goal of developing a synergy between expertise and competences, leverage female gender representation, and foster participation of young researchers.