  Ultracold atomic quantum gases belong to the most exciting challenges of
modern physics. Their theoretical description has drawn much from classical
field equations. These mean-field approximations are in general reliable for
dilute gases in which the atoms collide only rarely with each other, and for
situations where the gas is not too far from thermal equilibrium. With
present-day technology it is, however, possible to drive and observe a system
far away from equilibrium. Functional quantum field theory provides powerful
tools to achieve both, analytical understanding and numerical computability,
also in higher dimensions, of far-from-equilibrium quantum many-body dynamics.
In the article, an outline of these approaches is given, including methods
based on the two-particle irreducible effective action as well as on
renormalisation-group theory. Their relation to near-equilibrium kinetic theory
is discussed, and the distinction between quantum and classical statistical
fluctuations is shown to naturally emerge from the functional-integral
description. Example applications to the evolution of an ultracold atomic Bose
gas in one spatial dimension underline the power of the methods. The article is
compiled from the notes for lectures held at 46. Internationale
Universitaetswochen fuer Theoretische Physik 2008 in Schladming, Austria.
