1 The Geometry of tangent manifold 1.1 The manifold TM 1.2 Semisprays on the manifold TM 1.3 Nonlinear connections 1.4 N-linear connections 1.5 Parallelism. Structure equations 2 Lagrange spaces 2.1 The notion of Lagrange space 2.2 Variational problem. Euler-Lagrange equations 2.3 Canonical semispray. Nonlinear connection 2.4 Hamilton-Jacobi equations 2.5 Metrical N-linear connections 2.6 The electromagnetic and gravitational fields 2.7 The almost Kahlerian model of a Lagrange space Ln 2.8 Generalized Lagrange spaces 3 Finsler Spaces 3.1 Finsler metrics 3.2 Geodesics 3.3 Cartan nonlinear connection 3.4 Cartan metrical connection 4 The Geometry of Cotangent Manifold 4.1 Cotangent bundle 4.2 Variational problem. Hamilton-Jacobi equations 4.3 Nonlinear connections 4.4 N-linear connections 4.5 Parallelism, paths and structure equations 5 Hamilton spaces 5.1 Notion of Hamilton space 5.2 Nonlinear connection of a Hamilton space 5.3 The canonical metrical connection of Hamilton space Hn 5.4 Generalized Hamilton Spaces GHn 5.5 The almost Kahlerian model of a Hamilton space 6 Cartan spaces 6.1 Notion of Caftan space 6.2 Canonical nonlinear connection of Ln 6.3 Canonical metrical connection of Ln 6.4 The duality between Lagrange and Hamilton spaces 7 The Geometry of the manifold TkM 7.1 The bundle of acceleration of order k��1 7.2 The Liouville vector fields 7.3 Variational Problem 7.4 Semisprays. Nonlinear connections 7.5 The dual coefficients of a nonlinear connection 7.6 Prolongation to the manifold TkM of the Riemannian structures given on the base manifold M 7.7 N-linear connections on TkM 8 Lagrange Spaces of Higher-order 8.1 The spaces L(k)n = (M,L) 8.2 Examples of spaces L(k)n 8.3 Canonical metrical N-connection 8.4 The Riemannian (k - 1)n-contact model of the space L(k)n