These notes are based on lectures the author gave at the University of Bonn and the Erwin Schr?dinger Institute in Vienna. The aim is to give a thorough introduction to the theory of K?hler manifolds with special emphasis on the differential geometric side of K?hler geometry. The exposition starts with a short discussion of complex manifolds and holomorphic vector bundles and a detailed account of the basic differential geometric properties of K?hler manifolds. The more advanced topics are the cohomology of K?hler manifolds, Calabi conjecture, Gromov's K?hler hyperbolic spaces, and the Kodaira embedding theorem. Some familiarity with global analysis and partial differential equations is assumed, in particular in the part on the Calabi conjecture. There are appendices on Chern-Weil theory, symmetric spaces, and $L^2$-cohomology.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.