These notes are a survey of the material treated in a series of lectures 
delivered at the Xth Summer School {\em Jorge Andr\'e Swieca}. They are 
concerned with renormalization in Quantum Field Theories. At the level of 
perturbation series, we review classical results as Feynman graphs, ultraviolet 
and infrared divergences of Feynman integrals, Weinberg's theorem and Hepp's 
theorem,  the renormalization group and the Callan-Symanzik equation, the large 
order behavior and the divergence  of most perturbation series. Out of the 
perturbative regime, as an example of a constructive method, we review Borel 
summability and point out how it is possible to circumvent the perturbation 
diseases. These lectures are a preparation for the joint course given by 
Professor V. Rivasseau at the same school, where more sophisticated non-
perturbative analytical methods based on rigorous renormalization group 
techniques are presented, aiming at furthering our understanding about the 
subject and bringing field theoretical models to a satisfactory mathematical 
level.

Keywords: Mathematical Physics, Constructive Euclidean Quantum Field Theory, 
Renormalization, Renormalization Group.