We give in this chapter a brief description of Brownian motion and some associated processes: the running maximum and the running minimum, and regulators. We also discuss two of its extensions: Lévy processes and Markov-modulated Brownian motion (MMBM)—the topic of our book. We present a few basic properties, so as to render intuitively plausible, and wholly reasonable, the approach that we adopt throughout the book, that is, to connect physical properties of the various processes with the algebraic structures which we have to manipulate.