This course begins with an overview of measure theory, followed by topics that include random walks, the strong law of large numbers, the central limit theorem, martingales, Markov chains, characteristic functions, Poisson processes, and Brownian motion. Towards the end, some further topics may be covered, such as stochastic calculus, stochastic differential equations, Gaussian processes, random graphs, Markov chain mixing, random matrix theory, and interacting particle systems.