  Numerous correlated electron systems exhibit a strongly scale-dependent
behavior. Upon lowering the energy scale, collective phenomena, bound states,
and new effective degrees of freedom emerge. Typical examples include (i)
competing magnetic, charge, and pairing instabilities in two-dimensional
electron systems, (ii) the interplay of electronic excitations and order
parameter fluctuations near thermal and quantum phase transitions in metals,
(iii) correlation effects such as Luttinger liquid behavior and the Kondo
effect showing up in linear and non-equilibrium transport through quantum wires
and quantum dots. The functional renormalization group is a flexible and
unbiased tool for dealing with such scale-dependent behavior. Its starting
point is an exact functional flow equation, which yields the gradual evolution
from a microscopic model action to the final effective action as a function of
a continuously decreasing energy scale. Expanding in powers of the fields one
obtains an exact hierarchy of flow equations for vertex functions. Truncations
of this hierarchy have led to powerful new approximation schemes. This review
is a comprehensive introduction to the functional renormalization group method
for interacting Fermi systems. We present a self-contained derivation of the
exact flow equations and describe frequently used truncation schemes. Reviewing
selected applications we then show how approximations based on the functional
renormalization group can be fruitfully used to improve our understanding of
correlated fermion systems.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics;Superconductivity