In this chapter we describe the abstract definiton and the basic facts on algebraic Markov processes (see [5]). The main goal is to show that the fundamental definitions and properties of Markov processes are easiy formulated in an algebraic languaje suitable for the study of Markov processes appearing in quantum theory. Moreover, we discuss in detail the notion of complete positivity which turns out to be the natural generalisation of positivity for commutative (classical) case and a non-commutative version of the Feynman-Kac formula which is the basic ingredient in the construction of Markov cocycles and processes.