This compact yet thorough text zeros in on the parts of the theory that are useful for applications to mathematical finance, queuing theory, biology, and physics. It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential equations. It solves stochastic differential equations by a variety of methods and studies in detail the one dimensional case. This time-saving book concludes by treating semigroups and generators, applying the theory of Harris chains to diffusions, and presenting a quick course in weak convergence of Markov chains to diffusions.