I show that on a compact hyperbolic surface, the mass of an L2- normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassi- cal version, has many applications including control for the Schrdinger equation and the full support property for semiclassical defect mea- sures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.