The purpose of this survey paper is to examine some specific applications of the process of averaging over finite groups, compact topological groups, and compact homogeneous manifolds. These applications occur in a variety of different disciplines of pure and applied mathematics, such as (i) classical representation theory of finite groups, (ii) practical Fourier analysis, (iii) classical harmonic analysis, (iv) approximation theory, (v) complex analysis in several variables, and (vi) combinatorics. In each case, the symmetrization technique reveals itself to be a simple and efficient tool to produce far-reaching identities and inequalities in a unifying manner.