In the previous chapters we described the topological properties of circle diffeomorphisms, i.e. the order of the orbits, properties of the rotation number, non-existence of wandering intervals, density of the orbits, etc. We are now interested in studying the systems from a geometrical point of view. We learned that the orbits of diffeomorphisms are deformations of orbits of a rotation. Of course, on large scale we can deform the orbits in any possible way creating any kind of large scale geometry. For example, one can use a smooth conjugation to change the large scale geometrical properties. The natural question is then to ask whether it is possible to deform the geometry on small scales. A smooth conjugacy can not achieve that. For studying this question we need a microscope which allows us to look at the details of the dynamics at small scale. The mathematical object which will play the role of the microscope is the renormalization operator. In the following sections we introduce renormalization and discuss its properties. The geometrical use of renormalization will be discussed in the next chapter.