  The rich and diverse dynamics of particle-based systems ultimately originates
from the coupling of their degrees of freedom via internal interactions. To
arrive at a tractable approximation of such many-body problems, coarse-graining
is often an essential step. Power functional theory provides a unique and
microscopically sharp formulation of this concept. The approach is based on an
exact one-body variational principle to describe the dynamics of both
overdamped and inertial classical and quantum many-body systems. In
equilibrium, density functional theory is recovered, and hence spatially
inhomogeneous systems are described correctly. The dynamical theory operates on
the level of time-dependent one-body correlation functions. Two- and
higher-body correlation functions are accessible via the dynamical test
particle limit and the nonequilibrium Ornstein-Zernike route. We describe the
structure of this functional approach to many-body dynamics, including much
background as well as applications to a broad range of dynamical situations,
such as the van Hove function in liquids, flow in nonequilibrium steady states,
motility-induced phase separation of active Brownian particles, lane formation
in binary colloidal mixtures, and both steady and transient shear phenomena.
