  One of the important aspects in recent trends in complex analysis has been
the increasing degree of cross-fertilization between the latter and
mathematical physics with great benefits to both subjects. Contour dynamics in
the complex plane turned to be a meeting point for complex analysts,
specialists in stochastic processes, and mathematical physicists. This was
stimulated, first of all, by recent progress in understanding structures in the
classical and stochastic L\"owner evolutions, and in the Laplacian growth. The
Virasoro algebra provides a basic algebraic object in conformal field theory
(CFT) so it was not surprising that it turned to play an important role of a
structural skeleton for contour dynamics. The present paper is a survey of
recent progress in the study of the CFT viewpoint on contour dynamics, in
particular, we show how the Witt and Virasoro algebras are related with the
stochastic L\"owner and classical L\"owner-Kufarev equations.
