The Yau-Tian-Donaldson conjecture predicts that the existence of a canonical (i.e. Kähler-Einstein, constant scalar curvature, or more generally extremal) Kähler metric on a polarized complex manifold is governed by the algebro-geometric notion of K-stability. While the Kähler-Einstein case is by now completely understood, the general case remains open. However, tremendous progress has been accomplished, and the purpose of this talk is to review these recent developments.