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We survey some recent developments in the direction of the Yau-Tian-Donaldsonconjecture, which relates the existence of constant scalar curvature K\"ahlermetrics to the algebro-geometric notion of K-stability. The emphasis is put onthe use of pluripotential theory and the interpretation of K-stability in termsof non-Archimedean geometry.

Variational and non-Archimedean aspects of the Yau--Tian--Donaldson conjecture