Evolution of the concept known in the theoretical physics as the
Renormalization Group (RG) is presented. The corresponding symmetry, that has
been first introduced in QFT in mid-fifties, is a continuous symmetry of a
solution with respect to transformation involving parameters (e.g., of boundary
condition) specifying some particular solution.
After short detour into Wilson's discrete semi-group, we follow the expansion
of QFT RG and argue that the underlying transformation, being considered as a
reparameterisation one, is closely related to the self-similarity property. It
can be treated as its generalization, the Functional Self-similarity (FS).
Then, we review the essential progress during the last decade of the FS
concept in application to boundary value problem formulated in terms of
differential equations. A summary of a regular approach recently devised for
discovering the RG = FS symmetries with the help of the modern Lie group
analysis and some of its applications are given.
As a main physical illustration, we give application of new approach to
solution for a problem of self-focusing laser beam in a non-linear medium.