Let X be a complex projective manifold, and L an ample line bundle on X. The Yau--Tian--Donaldson conjecture predicts that the existence of a constant scalar curvature Kähler metric in the first Chern class of L is equivalent to a certain stability condition on (X,L) of algebro-geometric nature. I will report on joint work with S. Boucksom, where we settle this conjecture, with a stability condition inspired by non-Archimedean geometry.