Introduction. I. Elements of Coordinate-Free Differential Geometry. II. Introduction to Stochastic Analysis in Rn III. Stochastic Differential Equations on Manifolds. IV. Langevin's Equation in Geometric Form. V. Nelson's Stochastic Mechanics. VI. The Lagrangian Approach to Hydrodynamics. Appendix: Solution of the Newton-Nelson Equation with Random Initial Data Yu.E. Glikhlikh, T.J. Zastawniak. References. Index.