  The renormalization group plays an essential role in many areas of physics,
both conceptually and as a practical tool to determine the long-distance
low-energy properties of many systems on the one hand and on the other hand
search for viable ultraviolet completions in fundamental physics. It provides
us with a natural framework to study theoretical models where degrees of
freedom are correlated over long distances and that may exhibit very distinct
behavior on different energy scales. The nonperturbative functional
renormalization-group (FRG) approach is a modern implementation of Wilson's RG,
which allows one to set up nonperturbative approximation schemes that go beyond
the standard perturbative RG approaches. The FRG is based on an exact
functional flow equation of a coarse-grained effective action (or Gibbs free
energy in the language of statistical mechanics). We review the main
approximation schemes that are commonly used to solve this flow equation and
discuss applications in equilibrium and out-of-equilibrium statistical physics,
quantum many-particle systems, high-energy physics and quantum gravity.
