This text focuses on the parts of stochastic theory that are particularly relevant to applications. It begins with a description of Brownian motion and the associated stochastic calculus, including the relationship to partial differential equations. It then solves stochastic differential equations by a variety of methods. The author also studies in detail the one-dimensional case. The book concludes with a treatment of semigroups and generators, applying the theory of Harris chains to diffusions as well as weak convergence of Markov chains to diffusions.