  We briefly examine recent developments in the field of open quantum system
theory, devoted to the introduction of a satisfactory notion of memory for a
quantum dynamics. In particular, we will consider a possible formalization of
the notion of non-Markovian dynamics, as well as the construction of quantum
evolution equations featuring a memory kernel. Connections will be drawn to the
corresponding notions in the framework of classical stochastic processes, thus
pointing to the key differences between a quantum and classical formalization
of the notion of memory effects.
