The dynamical behavior of open quantum systems plays a key role in
many applications of quantum mechanics, examples ranging from
fundamental problems, such as the environment-induced decay of quantum
coherence and relaxation in many-body systems, to applications in
condensed matter theory, quantum transport, quantum chemistry and
quantum information. In close analogy to a classical Markovian
stochastic process, the interaction of an open quantum system with a
noisy environment is often modelled phenomenologically by means of a
dynamical semigroup with a corresponding time-independent generator in
Lindblad form, which describes a memoryless dynamics of the open system
typically leading to an irreversible loss of characteristic quantum
features. However, in many applications open systems exhibit pronounced
memory effects and a revival of genuine quantum properties such as
quantum coherence, correlations and entanglement. Here, recent
theoretical results on the rich non-Markovian quantum dynamics of open
systems are discussed, paying particular attention to the rigorous
mathematical definition, to the physical interpretation and
classification, as well as to the quantification of quantum memory
effects. The general theory is illustrated by a series of physical
examples. The analysis reveals that memory effects of the open system
dynamics reflect characteristic features of the environment which opens
a new perspective for applications, namely to exploit a small open
system as a quantum probe signifying nontrivial features of the
environment it is interacting with. This article further explores the
various physical sources of non-Markovian quantum dynamics, such as
structured environmental spectral densities, nonlocal correlations
between environmental degrees of freedom and correlations in the
initial system-environment state, in addition to developing schemes for
their local detection. Recent experiments addressing the detection,
quantification and control of non-Markovian quantum dynamics are also
briefly discussed.