  We provide a general construction of quantum generalized master equations
with memory kernel leading to well defined, that is completely positive and
trace preserving, time evolutions. The approach builds on an operator
generalization of memory kernels appearing in the description of non-Markovian
classical processes, and puts into evidence the non uniqueness of the
relationship arising due to the typical quantum issue of operator ordering. The
approach provides a physical interpretation of the structure of the kernels,
and its connection with the classical viewpoint allows for a trajectory
description of the dynamics. Previous apparently unrelated results are now
connected in a unified framework, which further allows to phenomenologically
construct a large class of non-Markovian evolutions taking as starting point
collections of time dependent maps and instantaneous transformations describing
the microscopic interaction dynamics.
