The renormalization group (RG) has nowadays achieved the status of a
meta-theory, which is a theory about theories. The theory of the RG
consists of a set of concepts and methods which can be used to
understand phenomena in many different ?elds of physics, ranging from
quantum ?eld theory over classical statistical mechanics to
nonequilibrium phenomena. RG methods are particularly useful to
understand phenomena where ?uctuations involving many different length
or time scales lead to the emergence of new collective behavior in
complex many-body systems. In view of the diversity of ?elds where RG
methods have been successfully applied, it is not surprising that a
variety of apparently different implementations of the RG idea have been
proposed. Unfortunately, this makes it somewhat dif?cult for beginners
to learn this technique. For example, the ?eld-theoretical formulation
of the RG idea looks at the ?rst sight rather different from the RG
approach pioneered by Wilson, the latter being based on the concept of
the effective action which is ite- tively calculated by successive
elimination of the high-energy degrees of freedom. Moreover, the
Wilsonian RG idea has been implemented in many different ways, depending
on the particular problem at hand, and there seems to be no canonical
way of setting up the RG procedure for a given problem.
