The central theme of the book is the fusion of two classical theories that were extensively developed in the course of the twentieth century, namely the Hamilton–Jacobi theory of separation of variables and Cartan geometry. The core concepts of the latter, notably the method of moving frames, are used to establish and develop the underlying ideas of Hamilton–Jacobi theory. The book is self-contained, beginning with an introduction to classical Riemannian geometry from Cartan's viewpoint aimed at developing the requisite background to be used in the study of Hamilton-Jacobi theory.As is well-known, the geometry of valence two Killing tensors plays pivotal role in the Hamilton–Jacobi theory of orthogonal separation of variables. From this perspective, much of the material presented in the book is devoted to the description of the geometry of valence two Killing tensors and orthogonal webs they define in pseudo–Riemannian spaces of constant curvature, using the concepts of isometry groups, principal fiber bundles, connections, orbit analysis, invariants, as well as other features of the moving frames method. An important feature of the book is that the fundamental concept of the Schouten bracket is used extensively to introduce and study the basic ideas of the classical Hamiltonian mechanics, where Hamilton–Jacobi theory is extensively used in integration of Hamiltonian systems.The book is suitable for a one-year graduate course in applicable differential geometry. It is well illustrated with numerous examples, applications and exercises. It will also be useful to experts in various areas of mathematical physics who are interested in applications of Cartan geometry.