  During the last ten years, the studies on non-Markovian open system dynamics
has become increasingly popular and having contributions from diverse set of
research communities. This interest has arisen due to fundamental problematics
how to define and quantify memory effects in the quantum domain, how to exploit
and develop applications based on them, and also due to the question what are
the ultimate limits for controlling open system dynamics. We give here a simple
theoretical introduction to the basic approaches to define and quantify quantum
non-Markovianity -- also highlighting their connections and differences. In
addition to the importance of the development for open quantum systems studies,
we also discuss the implications of the progress for other fields including,
e.g., formal studies of stochastic processes and quantum information science,
and conclude with possible future directions the recent developments open.
