The instability of anti-de Sitter space-time

In this review, we retrace the recent progress in the anti-de Sitter (AdS) instability problem. By instability we mean that for large classes of initial data, any perturbation of AdS space-time, however small, leads to the formation of a black hole. Since the seminal work of Bizo\'n and Rostworowski in 2011, many different kinds of numerical experiments were performed in asymptotically AdS space-times, unveiling a very intricate structure of the instability. In particular, many efforts were dedicated to the search of islands of stability, i.e.\ families of initial data that resist black hole formation. Many analytical and numerical tools were deployed to disentangle stable from unstable initial data, and shed new light on the necessary and sufficient conditions for collapse. Recently, research beyond spherical symmetry became more and more engaged. This is a very promising channel of investigation toward a deeper understanding of the gravitational dynamics in asymptotically AdS space-times.