The authors sought to create a tutorial regarding stochastic analysis related to Brownian motion, but the theme of the textbook became broader and more intense. The book starts with the introduction to stochastic processes including Kolmogorov consistency theory, trajectories and their smooth properties, measurability, stopping times, and some other basic concepts of stochastic processes. Further, the book examines the foundations of the Brownian motion smoothly passing with the help of the theory of martingales to analytic tools for Brownian motion. Such traditional elements of stochastic analysis as stochastic integration and stochastic differential equations are included and applied to martingale problems, the treatment of partial differential equations in probability and Gaussian solutions to stochastic equations. Jump Markov processes are studied as well as invariant measures for multidimensional diffusions and ergodic measures. The large deviations principle for diffusions concludes the book. The style of the book is clear and transparent, everything you need and nothing more. The book can be recommended for all specialists in probability and stochastic processes and its applications starting from the undergraduate and graduate students and ending with experienced professionals.