Stochastic optimization and online learning have played a vital role in machine learning where the full batch of data is either unavailable or too large to process in practice. This course introduces the theoretical foundations and algorithmic development in this area. The topics will start form the basic convex optimization theories as well as numerical methods, and we then focus on the stochastic approximation for stochastic optimization and online learning in many statistical and machine learning models, supplemented with the most recent progress from research literature. After this class, the students with some preliminaries of classic optimizations are expected to transit into the new optimization world in the machine learning, in which significant progresses have been made during the last decade.