  Semi-Markov processes represent a well known and widely used class of random
processes in classical probability theory. Here, we develop an extension of
this type of non-Markovian dynamics to the quantum regime. This extension is
demonstrated to yield quantum master equations with memory kernels which allow
the formulation of explicit conditions for the complete positivity of the
corresponding quantum dynamical maps, thus leading to important insights into
the structural characterization of the non-Markovian quantum dynamics of open
systems. Explicit examples are analyzed in detail.
