The Non-Linear Dynamics of Higher Dimensional Black Holes

In this talk I will review and discuss the progress made in recent years in the understanding of black holes and their dynamics beyond the traditional four-dimensional asymptotically flat setting, that is, in higher dimensions and in other asymptotics, e.g., Kaluza-Klein and anti-de Sitter. It turns out that in these situations, black hole instabilities are ubiquitous. Understading the non-linear evolution of such instabilities and their possible endpoints has revealed new gravitational phenomena and it can shed light to certain types of singularities that may form generically in general relativity.