This book presents the theory of complex Finsler geometry by integrating the traditional differential geometric approach with the analytic approach of complex analysis and the algebraic approach of algebraic geometry. Finsler geometry is treated as a special case of the geometry associated with the complex MongeAmpre equation. The theory of intrinsic metrics, especially the Kobayashi and the Caratheodory metrics, in complex analysis provides a wealth of important examples of complex Finsler manifolds. Furthermore, algebraic geometric techniques are very powerful in extending the classical Schwarz lemma, which is a highly indispensable tool in the classical theory of Finsler geometry and holomorphic curves, to more sophisticated and useful settings. This book gives a coherent account of the basic, fundamental and up-to-date modern theory of Finsler geometry that is easily accessible to beginning graduate students as well as researchers alike.