This book falls into two parts. The first studies those circle diffeomorphisms which commute with a diffeomorphism f whose rotation number r(f) is irrational. When f is not smoothly conjugated to a rotation, its rotation number r(f), according to Herman's theory, cannot be Diophantine. The centralizer, f, thus reflects how good are the rational and Diophantine approximations of r. The second part is dedicated to the study of biholomorphisms of a complex variable in the neighbourhood of a fixed point. In 1942 Siegel demonstrated that such a germ is analytically linearizable if its linear part is a Diophantine rotation; the arithmetic condition imposed on the rotation number was then weakened by Bruno. A geometrical approach to the problem yields an alternative proof of this result and also demonstrates the converse. If the Bruno's arithmetic condition is not satisfied the corresponding quadratic polynomial is not linearizable.