1 Introduction to Kรคhler manifolds -- 1.1 Kรคhler metrics -- 1.2 Curvature of Kรคhler metrics -- 2 Extremal Kรคhler metrics -- 2.1 The space of Kรคhler metrics -- 2.2 A brief review of Chern classes -- 2.3 Uniformization of Kรคhler-Einstein manifolds -- 3 Calabi-Futaki invariants -- 3.1 Definition of Calabi-Futaki invariants -- 3.2 Localization formula for Calabi-Futaki invariants -- 4 Scalar curvature as a moment map -- 5 Kรคhler-Einstein metrics with non-positive scalar curvature -- 5.1 The Calabi-Yau Theorem -- 5.2 Kรคhler-Einstein metrics for manifolds with c1(M) < 0 -- 6 Kรคhler-Einstein metrics with positive scalar curvature -- 6.1 A variational approach -- 6.2 Existence of Kรคhler-Einstein metrics -- 6.3 Examples -- 7 Applications and generalizations -- 7.1 A manifold without Kรคhler-Einstein metric -- 7.2 K-energy and metrics of constant scalar curvature -- 7.3 Relation to stability