This course offers an introduction to modern stochastic processes, including Markov processes, Poisson processes, Renewal processes, Martingales, Gaussian processes, and Brownian motion. The course will include not only a solid theoretical foundation, but also some applications to statistics and data science, including Queue theory, MCMC, reinforcement learning, and so on. This course is open to both high-level undergraduates and graduate students.  Upon successful completion, students will have the knowledge and skills to: 1. Explain the fundamental concepts of stochastic processes in both discrete time and continuous time. 2. Understand the position of stochastic processes in some modern applications in statistics and data science. 3. Apply problem-solving techniques using stochastic analysis methods in various situations. Besides these, graduate students should also be able to: Demonstrate mathematical reasoning through analyzing, proving, and explaining concepts from stochastic analysis.