  These notes are the second part of a common course on Renormalization Theory
given with Professor P. da Veiga at X Jorge Andre Swieca Summer School, Aguas
de Lindoia, Brazil, February 7-12, 1999. I emphasize the rigorous
non-perturbative or constructive aspects of the theory. The usual formalism for
the renormalization group in field theory or statistical mechanics is reviewed,
together with its limits. The constructive formalism is introduced step by
step. Taylor forest formulas allow to perform easily the cluster and Mayer
expansions which are needed for a single step of the renormalization group in
the case of Bosonic theories. The iteration of this single step leads to
further difficulties whose solution is briefly sketched. The second part of the
course is devoted to Fermionic models. These models are easier to treat on the
constructive level, so they are very well suited to beginners in constructive
theory. It is shown how the Taylor forest formulas allow to reorganize
perturbation theory nicely in order to construct the Gross-Neveu2 model without
any need for cluster or Mayer expansions. Finally applications of this
technique to condensed matter and renormalization group around Fermi surface
are briefly reviewed.
