  In these lectures we introduce the functional renormalization group out of
equilibrium. While in thermal equilibrium typically a Euclidean formulation is
adequate, nonequilibrium properties require real-time descriptions. For quantum
systems specified by a given density matrix at initial time, a generating
functional for real-time correlation functions can be written down using the
Schwinger-Keldysh closed time path. This can be used to construct a
nonequilibrium functional renormalization group along similar lines as for
Euclidean field theories in thermal equilibrium. Important differences include
the absence of a fluctuation-dissipation relation for general
out-of-equilibrium situations. The nonequilibrium renormalization group takes
on a particularly simple form at a fixed point, where the corresponding
scale-invariant system becomes independent of the details of the initial
density matrix. We discuss some basic examples, for which we derive a hierarchy
of fixed point solutions with increasing complexity from vacuum and thermal
equilibrium to nonequilibrium. The latter solutions are then associated to the
phenomenon of turbulence in quantum field theory.
