I will describe some of the ideas behind the proof of the global stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant. I will explain the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations, and thus how our solution scheme finds a suitable gauge, within a carefully chosen finite-dimensional family of gauges, in which we can find the global solution. I will also address the issue of finding the mass and the angular momentum of the final black hole. This talk is based on joint work with Andras Vasy.