this book presents the most recent advances in complex finsler geometry and related geometries: the geometry of complex lagrange, hamilton and cartan spaces. the last three spaces were initially introduced to and have been investigated by the author of the present volume over the past several years. this book will acquaint the reader with: - a survey of some basic results from complex manifolds and the complex vector bundles theory, - the geometry of holomorphic tangent bundles, - an analysis of the main results in complex finsler geometry, - a study of the geometry of complex lagrange and generalized lagrange spaces. of special interest are their holomorphic subspaces, - the construction of the complex hamilton geometry, - the complex finsler vector bundles. audience: geometers, complex analysts, and physicists in quantum field theory and in theoretical mechanics will find this book of interest. the volume can be also used as a supplementary graduate text.