The discovery of the Kerr solution in 1963 has raised the possibility that there exist astrophysical objects in the Universe to which these pure mathematical objects- stationary solutions of the Einstein equations in vacuum- correspond to, that is rotating black holes. For  that to happen it was necessary to show that these new solutions are stable. That has led to the well known Kerr stability conjecture, and  a sustained effort by both physicists and mathematicians to establish its validity. The conjecture has been now verified in the particular case of small angular momentum.

 The full proof of the nonlinear stability of Kerr consists of five papers, three written in collaboration with Jeremie Szeftel, one in collaboration with Elena Giorgi and Jeremie Szeftel and another supporting paper authored by Dawei Shen. In my lecture I will state the main conjecture, give a short review of previous developments in the field, describe the main architecture of the new result as well as some of the most important new ideas behind the proof.