The renormalization operator was introduced to study the small scale properties of generalized circle diffeomorphisms. It turns out that on small scale, these maps behave like rotations. In the sense that consecutive renormalizations converge to rotations. This central and impressive aspect of renormalization theory implies that the geometrical aspects of orbits on small scales are determined by the rotations. The following three chapters are dedicated to explaining this convergence phenomenon. First, in this chapter, we discuss the three ingredients which are responsible for the convergence of renormalization to the rotations. There is a specific mechanism which is responsible for each of the following aspects of convergence.