Local Theory -- Complex Manifolds -- Kähler Manifolds -- Vector Bundles -- Applications of Cohomology -- Deformations of Complex Structures.Complex geometry studies (compact) complex manifolds. It discusses algebraic as well as metric aspects. The subject is on the crossroad of algebraic and differential geometry. Recent developments in string theory have made it an highly attractive area, both for mathematicians and theoretical physicists. The author’s goal is to provide an easily accessible introduction to the subject. The book contains detailed accounts of the basic concepts and the many exercises illustrate the theory. Appendices to various chapters allow an outlook to recent research directions. Daniel Huybrechts is currently Professor of Mathematics at the University Denis Diderot in Paris.