  We construct a large class of non-Markovian master equations that describe
the dynamics of open quantum systems featuring strong memory effects, which
relies on a quantum generalization of the concept of classical semi-Markov
processes. General conditions for the complete positivity of the corresponding
quantum dynamical maps are formulated. The resulting non-Markovian quantum
processes allow the treatment of a variety of physical systems, as is
illustrated by means of various examples and applications, including quantum
optical systems and models of quantum transport.
