This talk was part of the Workshop on "Mathematical Relativity: Past, Present, Future" held at the ESI December 4 — 7, 2023. I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.