Primary subject categories: • Probability theory and stochastic processes • Brownian motionSecondary subject categories: • Stochastic integrals • Stochastic ordinary differential equations (aspects of stochastic analysis) • Transition functions, generators and resolvents • Martingales and classical analysis • Diffusion processes • Continuous-time Markov processes on general state spacesGeneral textbooks on probability theory cover, if at all, Brownian motion only briefly. On the other hand, there is a quite substantial gap to more specialized texts on Brownian motion which is not so easy to overcome for the novice. Our aim was to write a book which can be used in the classroom as an introduction to Brownian motion and stochastic calculus, and as a first course in continuous-time and continuous-state Markov processes. My aim was to have a text which is both a self-contained back-up and self-study text for contemporary applications (such as mathematical finance) and a foundation to more advanced monographs, e.g. Ikeda & Watanabe [113], Revuz & Yor [216] or Rogers & Williams [222] (for stochastic calculus), Marcus & Rosen [178] (for Gaussian processes), Peres & Mörters [186] (for random fractals), Chung [33] or Port & Stone [209] (for potential theory) or Blumenthal & Getoor [18] (for Markov processes) to name but a few.