This course is a measure-theoretic introduction to the theory of continuous-time stochastic processes. We intend to treat some classical, fundamental results and give an overview of two important classes of processes. These processes are so-called martingales and Markov processes. The main part of the course is devoted to developing fundamental results in martingale theory and Markov process theory, with an emphasis on the interplay between the two worlds. The general results will then be used to study the fascinating properties of Brownian motion, an important process that is both a martingale and a Markov process. We also plan to study some applications in queueing theory.