In 2013 Martin Hairer published his theory of regularity structures. Drawing upon rough paths for inspiration this theory generalized the notion of "taylor expansion" and classical notions of regularity. Moreover, this theory had the flexibility to allow one to incorporate renormalization - essential for the treatment of singular SPDE.

In the years since then our understanding of how to implement renormalization in regularity structures has developed rapidly. I will sketch the overall framework of the theory of regularity structures and then summarize these recent developments in order to give an idea of the current state of the theory - this will draw upon the work of Martin Hairer, Lorenzo Zambotti, Yvan Bruned, llya Chevyrev, and the speaker.