Critical phenomena arise in a wide variety of physical systems. Classi-
cal examples are the liquid-vapour critical point or the paramagnetic-
ferromagnetic transition. Further examples include multicomponent fluids
and alloys, superfluids, superconductors, polymers and fully developed
tur- bulence and may even extend to the quark-gluon plasma and the early
uni- verse as a whole. Early theoretical investigators tried to reduce
the problem to a very small number of degrees of freedom, such as the
van der Waals equation and mean field approximations, culminating in
Landau's general theory of critical phenomena. Nowadays, it is
understood that the common ground for all these phenomena lies in the
presence of strong fluctuations of infinitely many coupled variables.
This was made explicit first through the exact solution of the
two-dimensional Ising model by Onsager. Systematic subsequent
developments have been leading to the scaling theories of critical
phenomena and the renormalization group which allow a precise
description of the close neighborhood of the critical point, often in
good agreement with experiments. In contrast to the general
understanding a century ago, the presence of fluctuations on all length
scales at a critical point is emphasized today. This can be briefly
summarized by saying that at a critical point a system is scale
invariant. In addition, conformal invaTiance permits also a non-uniform,
local rescal- ing, provided only that angles remain unchanged.
