Motivated in part by the wish to find interesting large distance modifications of gravity to be useful in cosmology, a simple idea is to try to give a mass to the graviton. The linear theory for such a "massive gravity" is known for a long time as the Fierz-Pauli theory. However, it does not recover predictions from linearized General Relativity in the massless limit (this is the celebrated "van Dam-Veltman-Zakharov - vDVZ - discontinuity"). A way out was proposed in 1972 by A. Vainshtein. This "Vainshtein mechanism" relies on non linearities which in turn lead to other pathologies which for a long time were thought to be unavoidable in any theory with a finite number of massive graviton. Following in particular developments links to brane worlds models such as the DGP model (which has some relation to "massive gravity") various issues related to pathologies of massive gravity are now better understood. I will review the pathologies of massive gravity and their possible cures, insisting in particular on the Vainshtein mechanism, as well as on the recent construction of de Rham-Gabadadze-Tolley.