Explore a comprehensive lecture on canonical Kähler metrics and the stability of algebraic varieties. Delve into topics such as Kähler manifolds, projective manifolds, holomorphic sectional curvature, and Ricci curvature. Examine constant scalar curvature Kähler metrics and the Yau-Tian-Donaldson conjecture. Investigate recent progress in the field, including the Kähler-Einstein case and the space of Kähler metrics. Study energy functionals, pluripotential theory, and variational approaches. Learn about test configurations, algebraic functionals, and uniform K-stability. Explore the K-stability of Fano varieties and the relationship between analytic and algebraic invariants. Discover approximation approaches to the Yau-Tian-Donaldson conjecture and the concept of K-stability over models.