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K-stability was first defined in complex geometry by Tian in late 90s and then reformulated by Donaldson in algebraic terms, to characterize the existence of Kähler-Einstein metrics on Fano varieties. In the last decade, a purely algebro-geometric theory has been developed. The theory combines deep techniques in higher dimensional algebraic geometry, with a circle of new perspectives from K-stability theory. Major outputs then include a moduli theory for Fano varieties, a new stability theory of singularities, as well as many new examples of Kähler-Einstein Fano varieties etc.