  Classical dynamical density functional theory (DDFT) is one of the
cornerstones of modern statistical mechanics. It is an extension of the highly
successful method of classical density functional theory (DFT) to
nonequilibrium systems. Originally developed for the treatment of simple and
complex fluids, DDFT is now applied in fields as diverse as hydrodynamics,
materials science, chemistry, biology, and plasma physics. In this review, we
give a broad overview over classical DDFT. We explain its theoretical
foundations and the ways in which it can be derived. The relations between the
different forms of deterministic and stochastic DDFT as well as between DDFT
and related theories, such as quantum-mechanical time-dependent DFT, mode
coupling theory, and phase field crystal models, are clarified. Moreover, we
discuss the wide spectrum of extensions of DDFT, which covers methods with
additional order parameters (like extended DDFT), exact approaches (like power
functional theory), and systems with more complex dynamics (like active
matter). Finally, the large variety of applications, ranging from fluid
mechanics and polymer physics to solidification, pattern formation, biophysics,
and electrochemistry, is presented.
