As an important branch of probability theory, stochastic process is a mathematical tool for studying the regularity of random phenomena over time and an important part of the random series of courses. It comes from reality and has a profound application background. It can be widely used in finance, economics and management science, information science, biological science, computer science and other engineering technology fields. The stochastic process itself is also an important foundation for studying stochastic analysis and mathematical finance in the future. This course starts with the martingale. We introduce the definition of martingale, stopping times and the martingale convergence theorems. Next, we study the definitions of Markov chain, and introduce the invariant measure and ergodic theorems. Finally, we contruct the Brownian motion, introduce the Wiener measure and discuss the applications of Brownian motion in partial differential equations. After completing this course, students should understand and master the basic concepts and conclusions of stochastic processes; master the definitions of martigale and stopping times, and the associated convergence theorems; master the definition and properties of Markov processes; master the definitions and properties of the Brownian motion and understand the wide applications of Brownian motion in modern probability theory.