This is an introduction to a new field in applied group analysis. Namely, the hook deals with the so-called renormalization group (briefly renormgroup) symmetries considered in the framework of approximate transformation groups. The notion of the renormalization group and the renormalization group method were introduced in theoretical physics by N. N. Bogoliubov and D. V. Shirkov in 1950s. Renormgroup symmetries provide a basis for the renormgroup algorithm for improving solutions to boundary value problems by converting "less applicable solutions" into "more applicable solutions". The algorithm is particularly useful for improving approximate solutions given by the perturbation theory.