Complex Analytic Geometry is one of the most important fields of Mathematics. It has a long history that culminated in the Cauchy integral formula in the 19th century. The theory was vastly developed and closely related to many other fields of Mathematics as well as the Sciences, where numerous applications have been found.This book starts off with the basic material, introducing characteristic classes mainly via the Chern–Weil theory, explaining the idea of localization of characteristic classes and presenting the aforementioned invariants and relations in a unified way from this perspective. Its exposition is carried out in a self-containing manner. Recent developments are also discussed.The profound consequences of this subject will make the book useful for mathematical students on fields as diverse as Algebraic Geometry, Differential Geometry, Topology, Complex Dynamical Systems and Mathematical Physics.