This monograph presents for the first time the foundations of Hamilton Geometry. The concept of Hamilton Space, introduced by the first author and investigated by the authors, opens a new domain in differential geometry with large applications in mechanics, physics, optimal control, etc. The book consists of thirteen chapters. The first three chapters present the topics of the tangent bundle geometry, Finsler and Lagrange spaces. Chapters 4-7 are devoted to the construction of geometry of Hamilton spaces and the duality between these spaces and Lagrange spaces. The dual of a Finsler space is a Cartan space. Even this notion is completely new, its geometry has the same symmetry and beauty as that of Finsler spaces. Chapter 8 deals with symplectic transformations of cotangent bundle. The last five chapters present, for the first time, the geometrical theory and applications of Higher-Order Hamilton spaces. In particular, the case of order two is presented in detail. Audience: mathematicians, geometers, physicists, and mechanicians. This volume can also be recommended as a supplementary graduate text.