We present a comprehensive and up to date review on the concept of quantum
non-Markovianity, a central theme in the theory of open quantum systems. We
introduce the concept of quantum Markovian process as a generalization of the
classical definition of Markovianity via the so-called divisibility property
and relate this notion to the intuitive idea that links non-Markovianity with
the persistence of memory effects. A detailed comparison with other definitions
presented in the literature is provided. We then discuss several existing
proposals to quantify the degree of non-Markovianity of quantum dynamics and to
witness non-Markovian behavior, the latter providing sufficient conditions to
detect deviations from strict Markovianity. Finally, we conclude by enumerating
some timely open problems in the field and provide an outlook on possible
research directions.Comment: Review article. Close to published versio
