The Einstein-Cartan-Sciama-Kibble theory of gravity naturally extends general relativity to account for the intrinsic spin of elementary particles, which causes spacetime to exhibit a geometric property called torsion. In fermionic spin-fluid matter, at densities much larger than the nuclear density, torsion manifests itself as a force that counters gravity. Such a gravitational repulsion not only prevents the formation of singularities in collapsing black holes, but also allows for a scenario in which every black hole produces a new universe inside its event horizon. Accordingly, our own Universe may be the interior of a black hole existing in another universe, with the Big Bang being replaced by a "Big Bounce". This scenario explains why the Universe today appears spatially flat, homogeneous and isotropic, without needing cosmic inflation. It also explains the arrow of time. Furthermore, if we were living in a rotating black hole then our Universe should have a "preferred direction" which could be related to the observed asymmetry in neutrino and neutral-meson oscillations. Finally, torsion generates in the Dirac equation a nonlinear-spinor term that could be the source of the observed matter-antimatter imbalance and of the cosmological constant. This term may also provide a self-regulated ultraviolet behavior of Dirac particles in quantum field theory. Torsion may be therefore a unifying concept of gravitational physics that could solve several major problems in cosmology and particle physics.