In this talk we discuss stability of holomorphic dynamical systems under perturbation. In dimension 1, the theory is now classical and is based on works by Lyubich, Mané-Sad-Sullivan and DeMarco. Starting from the basics, I will review this theory and (maybe) present a recent generalisation valid in any dimension. I will mainly focus on the arguments that do not readily generalise to this setting, and on the tools and ideas that allow one to overcome these problems.