The course mainly develops and applies probability theory to random events evolving over time. The tools developed in the course have been widely used in many disciplines, including physics, computer science, and finance. In particular, the use of stochastic processes is indispensable in the fields of artificial intelligence, machine learning and data science. In this course, we will systematically learn the basics of stochastic processes, including Markov chains, Poisson processes, Renewal processes, Martingales and Brownian motion. We will demonstrate how to use these probabilistic models to model real-world problems by studying many examples, with emphasis on applications in data science. The main focus of the course is to explain how to apply probabilistic tools to rigorously analyze these models.