  We review the most recent developments in the theory of open quantum systems
focusing on situations in which the reservoir memory effects, due to
long-lasting and non-negligible correlations between system and environment,
play a crucial role. These systems are often referred to as non-Markovian
systems. After a brief summary of different measures of non-Markovianity that
have been introduced over the last few years we restrict our analysis to the
investigation of information flow between system and environment. Within this
framework we introduce an important application of non-Markovianity, namely its
use as a quantum probe of complex quantum systems. To illustrate this point we
consider quantum probes of ultracold gases, spin chains, and trapped ion
crystals and show how properties of these systems can be extracted by means of
non-Markovianity measures.
