The purpose of the manual is to present the foundations of both the classical and modern theory of stochastic differential equations (SDE) and its connections with the theory of (linear and non-linear) partial differential equations and systems of such equations that arise in various applications. Particular attention is paid to the construction of probabilistic representations of solutions to the Cauchy problem for nonlinear parabolic equations and systems, which make it possible to reduce the parabolic problem to solving the corresponding SDEs and calculating the average of their functionals. The last two chapters are devoted to applications to problems of mathematical physics and financial mathematics.