1. General properties of nonlinear dynamical systems. 1.1. Finite-dimensional dynamical systems. 1.2. Poissonian and symplectic structures on manifolds -- 2. Nonlinear dynamical systems with symmetry. 2.1. The Poisson structures and Lie group actions on manifolds : Introduction. 2.2. Lie group actions on Poisson manifolds and the orbit structure. 2.3. The canonical reduction method on symplectic spaces and related geometric structures on principal fiber bundles. 2.4. The form of reduced symplectic structures on cotangent spaces to Lie group manifolds and associated canonical connections. 2.5. The geometric structure of abelian Yang-Mills type gauge field equations via the reduction method. 2.6. The geometric structure of non-abelian Yang-Mills gauge field equations via the reduction method. 2.7. Classical and quantum integrability -- 3. Integrability by quadratures. 3.1. Introduction. 3.2. Preliminaries. 3.3. Integral submanifold embedding problem for an abelian Lie algebra of invariants. 3.4. Integral submanifold embedding problem for a nonabelian Lie algebra of invariants. 3.5. Examples. 3.6. Existence problem for a global set of invariants. 3.7. Additional examples -- 4. Infinite-dimensional dynamical systems. 4.1. Preliminary remarks. 4.2. Implectic operators and dynamical systems. 4.3. Symmetry properties and recursion operators. 4.4. Backlund transformations. 4.5. Properties of solutions of some infinite sequences of dynamical systems. 4.6. Integro-differential systems -- 5. Integrability : The gradient-holonomic algorithm. 5.1. The Lax representation. 5.2. Recursive operators and conserved quantities. 5.3. Existence criteria for a Lax representation. 5.4. The current Lie algebra on a cycle : A symmetry subalgebra of compatible bi-Hamiltonian nonlinear dynamical systems -- 6. Algebraic, differential and geometric aspects of integrability. 6.1. A non-isospectrally Lax integrable KdV dynamical system. 6.2. Algebraic structure of the gradient-holonomic algorithm for Lax integrable systems.