Symmetry considerations are essential in many successful physical theories, for instance the Standard Model of Particle Physics, to characterize phases of matter, and so on. In General Relativity, different, but related, notions of symmetry arise. For instance, invariance under coordinate transformations underlies the geometric nature of the theory, asymptotic symmetries characterize isolated systems and their excitations such as gravitational waves, isometries characterize preferred equilibrium states such as Kerr black holes or deSitter space, while "hidden" symmetries can be used to generate complicated solutions. These symmetries, and the breaking thereof, are related in a deep way to the dynamics of General Relativity and to each other. In this talk, I outline some of these connections.