  The basic features of the dynamics of open quantum systems, such as the
dissipation of energy, the decay of coherences, the relaxation to an
equilibrium or non-equilibrium stationary state, and the transport of
excitations in complex structures are of central importance in many
applications of quantum mechanics. The theoretical description, analysis and
control of non-Markovian quantum processes play an important role in this
context. While in a Markovian process an open system irretrievably loses
information to its surroundings, non-Markovian processes feature a flow of
information from the environment back to the open system, which implies the
presence of memory effects and represents the key property of non-Markovian
quantum behavior. Here, we review recent ideas developing a general
mathematical definition for non-Markoviantiy in the quantum regime and a
measure for the degree of memory effects in the dynamics of open systems which
are based on the exchange of information between system and environment. We
further study the dynamical effects induced by the presence of
system-environment correlations in the total initial state and design suitable
methods to detect such correlations through local measurements on the open
system.
