1. Preliminary Considerations -- 2. Elements of Differential Calculus for Tensor Fields -- 2.1 Tensors -- 2.2 Tensor Fields -- 2.3 Transformation Properties of Tensor Fields -- 2.4 Directional Derivative of Tensor Fields -- 2.5 Differential ?-Forms -- 2.6 Flows and Lie Transport -- 2.7 Lie Derivatives -- 3. The Theory of Hamiltonian and Bi-Hamiltonian Systems -- 3.1 Lie Algebras -- 3.2 Hamiltonian and Bi-Hamiltonian Vector Fields -- 3.3 Symmetries and Conserved Quantities of Dynamical Systems -- 3.4 Tensor Invariants of Dynamical Systems -- 3.5 Algebraic Properties of Tensor Invariants -- 3.6 The Miura Transformation -- 4. Lax Representations of Multi-Hamiltonian Systems -- 4.1 Lax Operators and Their Spectral Deformations -- 4.2 Lax Representations of Isospectral and Nonisospectral Hierarchies -- 4.3 The Lax Operator Algebra -- 5. Soliton Particles -- 5.1 General Aspects -- 5.2 Algebraic Structure of Linear Systems -- 5.3 Algebraic Structure of Multi-Soliton Representation -- 5.4 Multi-Soliton Perturbation Theory -- 6. Multi-Hamiltonian Finite Dimensional Systems -- 6.1 Stationary Flows of Infinite Systems. Ostrogradsky Parametrizations -- 6.2 Stationary Flows of Infinite Systems. Newton Parametrization -- 6.3 Constrained Flows of Lax Equations -- 6.4 Restricted Flows of Infinite Systems -- 6.5 Separability of Bi-Hamiltonian Chains with Degenerate Poisson Structures -- 6.6 Nonstandard Multi-Hamiltonian Structures and Their Finite Dimensional Reductions -- 6.7 Bi-Hamiltonian Chains on Poisson-Nijenhuis Manifolds -- 7. Multi-Hamiltonian Lax Dynamics in (1+1)-Dimensions -- 7.1 Hamiltonian Dynamics on Lie Algebras -- 7.2 Basic Facts About R-Structures -- 7.3 Multi-Hamiltonian Dynamics of Pseudo-Differential Lax Operators -- 7.4 Multi-Hamiltonian Dynamics of Shift Lax Operators -- 8. Towards a Multi-Hamiltonian Theory of (2+1)-Dimensional Field Systems -- 8.1 The Sato Theory -- 8.2 Multi-Hamiltonian Lax Dynamics for Noncommutative Variables -- References.