  Quantum theory provides an extensive framework for the description of the
equilibrium properties of quantum matter. Yet experiments in quantum simulators
have now opened up a route towards generating quantum states beyond this
equilibrium paradigm. While these states promise to show properties not
constrained by equilibrium principles such as the equal a priori probability of
the microcanonical ensemble, identifying general properties of nonequilibrium
quantum dynamics remains a major challenge especially in view of the lack of
conventional concepts such as free energies. The theory of dynamical quantum
phase transitions attempts to identify such general principles by lifting the
concept of phase transitions to coherent quantum real-time evolution. This
review provides a pedagogical introduction to this field. Starting from the
general setting of nonequilibrium dynamics in closed quantum many-body systems,
we give the definition of dynamical quantum phase transitions as phase
transitions in time with physical quantities becoming nonanalytic at critical
times. We summarize the achieved theoretical advances as well as the first
experimental observations, and furthermore provide an outlook onto major open
questions as well as future directions of research.
