We consider the KPZ equation (or after a Cole-Hopf transform, SHE, or nonlinear variants of the latter) in one dimension driven by noise that is rougher than white, e.g. it is derivative of white of order gamma. It is known that a change in behavior takes place at gamma=1/4, as opposed to the criticality threshold gamma=1/2. We describe some recent contributions in both the gamma<1/4 and gamma>1/4 regimes, complementing works of [Hu-Huang-Le-Nualart-Tindel '17] and [Hairer '24]. Based partly on joint work with Fabio Toninelli.