  In the last 50 years, equilibrium density functional theory (DFT) has been
proven to be a powerful, versatile and predictive approach for the statics and
structure of classical particles. This theory can be extended to the
nonequilibrium dynamics of completely overdamped Brownian colloidal particles
towards so-called dynamical density functional theory (DDFT). The success of
DDFT makes it a promising candidate for a first-principle description of active
matter. In this lecture, we shall first recapitulate classical DDFT for passive
colloidal particles typically described by Smoluchowski equation. After a basic
derivation of DDFT from the Smoluchowski equation, we discuss orientational
degrees of freedom and the effect of hydrodynamic interactions for passive
particles. This brings us into an ideal position to generalize DDFT towards
active matter. In particular we distinguish between "dry active matter" which
is composed of self-propelled particles that contain no hydrodynamic flow
effects of a surrounding solvent and "wet active matter" where the hydrodynamic
flow fields generated by the microswimmers are taken into account. For the
latter, DDFT is a tool which unifies thermal fluctuations, direct particle
interactions, external driving fields and hydrodynamic effects arising from
internal self-propulsion discriminating between "pushers" and "pullers". A
number of recent applications is discussed including transient clustering of
self-propelled rods and the spontaneous formation of a hydrodynamic pump in
confined microswimmers.
