Abstract: The renormalization scheme is an essential tool for understanding polygonal exchange transformations (PETs), which are two-dimensional analogues for interval exchange transformations (IETs). We first review how the renormalization procedure works out in the IET case. Then, we consider several classes of examples of PETs, many of which arise from the study of polygonal outer billiards. In various cases, it turns out that there is a "renormalization map" defined on the set of parameters indexing PETs and for parameter values that are suitably "quadratic", there is a self-renormalization of the corresponding dynamical systems. Our examples will be taken from works of S. Tabachnikov, R. Schwartz, P. Hooper, and others.