I. Finite-Dimensional Differential Geometry and Mechanics. - 1 Some Geometric Constructions in Calculus on Manifolds. - 2 Geometric Formalism of Newtonian Mechanics. - 3 Accessible Points of Mechanical Systems. - II. Stochastic Differential Geometry and its Applications to Physics. - 4 Stochastic Differential Equations on Riemannian Manifolds. - 5 The Langevin Equation. - 6 Mean Derivatives Nelson's Stochastic Mechanics and Quantization. - III. Infinite-Dimensional Differential Geometry and Hydrodynamics. - 7 Geometry of Manifolds of Diffeomorphisms. - 8 Lagrangian Formalism of Hydrodynamics of an Ideal Incompressible Fluid. - 9 Hydrodynamics of a Viscous Incompressible Fluid and Stochastic Differential Geometry of Groups of Diffeomorphisms. - Appendices. - A. Introduction to the Theory of Connections. - Connections on Principal Bundles. - Connections on the Tangent Bundle. - Covariant Derivatives. - Connection Coefficients and Christoffel Symbols. - Second-Order Differential Equations and the Spray. - The Exponential Map and Normal Charts. - B. Introduction to the Theory of Set-Valued Maps. - C. Basic Definitions of Probability Theory and the Theory of Stochastic Processes. - Stochastic Processes and Cylinder Sets. - The Conditional Expectation. - Markovian Processes. - Martingales and Semimartingales. - D. The Itô Group and the Principal Itô Bundle. - E. Sobolev Spaces. - F. Accessible Points and Closed Trajectories of Mechanical Systems (by Viktor L. Ginzburg). - Growth of the Force Field and Accessible Points. - Accessible Points in Systems with Constraints. - Closed Trajectories of Mechanical Systems. - References. Language: English