Abstract: In contrast with the classical stability of stationary, asymptotically flat black holes in four dimensions, some families of higher dimensional black holes suffer from dynamical instabilities. By presenting a mathematical theorem on five-dimensional black rings recently proven in my doctoral thesis, I will argue for the existence of a new instability affecting not only black rings, but a wide class of higher dimensional black holes. This new instability can be directly related to a geometric property of higher dimensional black holes and should be seen as the manifestation of a more general dispersive phenomenon independent of the global, algebraic nature of the spacetime considered.