The classical way of measuring the regularity of a function is by comparing it in the neighbourhood of any point with a polynomial of sufficiently high degree. Would it be possible to replace monomials by functions with less regular behaviour or even by distributions? It turns out that the answer to this question has surprisingly far-reaching consequences for building solution theories for semilinear PDEs with very rough input signals, revisiting the age-old problem of multiplying distributions of negative order, and understanding renormalisation theory. As an application, we build the natural "Langevin equation" associated with Phi^4 Euclidean quantum field theory in dimension 3.