Abstract: We discuss a new theory of gravity, recently proposed by Banados and Ferreira. This theory is inspired by an original proposal by Eddington and it is equivalent to General Relativity in vacuum. However, it dramatically differs from Eistein's gravity inside matter. This "Eddington-inspired" gravity can be seen as a prototype of alternative theories that modify the coupling to matter in Einstein's equations. We show that this viable, one parameter theory may avoid Big-Bang singularities in early cosmology and may resolve singularities in the stellar collapse. Furthermore, the theory supports stable, compact pressureless stars made of perfect fluid, which provide interesting models of self-gravitating dark matter. Finally, we discuss possible observational constraints coming from solar physics.