This chapter introduces an axiomatic formulation of perturbative quantum field theory based on Wilsonian renormalization and effective field theories. The key points are:

1) Quantum field theories are defined as Wilsonian low-energy theories on Euclidean space or manifolds, constructed from functionals depending on background fields. 

2) The main theorem states that any quantum field theory constructed in this way can be defined by a local Lagrangian with only finitely many terms, up to renormalization. 

3) Renormalizability is proven by constructing local counterterms order-by-order in perturbation theory to cancel singularities arising in Feynman diagrams.

(Mathematical Surveys and Monographs 170) Kevin Costello - Renormalization and Effective Field Theory-American Mathematical Society (2011) PDF

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