I will discuss some recent work on a priori bounds for SDEs and SPDES that crucially leverages an algebraic insight into the regularity condition imposed on modelled distributions in the theory of regularity structures. This is based on one joint work with Augustin Moinat and Hendrik Weber and another joint work with Timothee Bonnefoi, Augusin Moinat, and Hendrik Weber. No prior knowledge of regularity structures is needed for this talk.