This is a status report on our on-going research on alternative theories of gravity and their astrophysical imprints. During the last century, General Relativity (GR) passed many stringent tests, and is now accepted as the standard theory of gravity. Nevertheless, most experiments can only probe the weak-field (low-curvature) regime, while the strong curvature regime remains essentially unexplored. In this regime the dynamics of black holes and neutron stars can sensibly differ from GR, with potentially observable effects. In the last decades several new alternative theories have been proposed. They pass all Solar System tests, but differ from GR in the strong curvature regime, for example they allow for different BH solutions and new classes of compact stars. These theories are viable candidates and should be taken as serious alternatives to the GR paradigm. We give an overview on strong curvature effects around black holes and compact stars in some alternative theory, discussing some results, limitations, future developments and prospects.