We discuss product theorems in groups acting on hyperbolic spaces: for every hyperbolic group there exists a constant a > 0 such that for every finite subset U that is not contained in a virtually cyclic subgroup, | U3 | > ( a | U | )2 . We also discuss the growth of | U^n | and conclude that the entropy of U (the limit of 1/n log | U^n | as n goes to infinity) exceeds 1/2 log ( a | U | ) . This generalizes results of Razborov and Safin, and answers a question of Button. We discuss similar estimates for groups acting acylindrically on trees or hyperbolic spaces. This talk is on a joint work with T. Delzant.