The stochastic quantization equation is a simple model for the kind of problems linked to locality and non-triviality of quantum field theories. In this talk we review recent advances in undestanding of the functional analytic structure of solutions to non-linear SPDEs and their application to the study of the stochastic quantization of a scalar field in 3 dimensions. These advances have been possible thanks to a generalization of the theory of (controlled) rough paths which allows a pathwise formulation to stochastic differential equations driven by irregular signal. In particular we discuss the role of multiscale decomposition of distributions and of the notion of paraproduct in the analysis of this problem.