Locality, the beta function and the Dynkin idempotent

In this talk we review briefly Connes-Kreimer's Hopf algebra approach to perturbative renormalization. We present direct proofs of the main combinatorial properties of the renormalization procedures, using the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and Hopf algebras and their associated descent algebras. This joint work with F. Patras and J.M. Gracia-Bondia.