We review some of our recent work on the infrared subtraction problem beyond NLO, focusing in particular on the general definition of local subtraction counterterms. Our approach takes as a starting point the infrared factorisation of virtual corrections to fixed-angle scattering amplitudes in massless gauge theories, and uses that structure to identify a set of matrix elements of field operators and Wilson lines, which can be used as local soft and collinear counterterms to all orders in perturbation theory. Tracing the connection between factorisation and subtraction uncovers significant simplifications in the structure of counterterms, especially for nested singular regions, and we believe that it will be a crucial ingredient for the construction of minimal, stable, and efficient subtraction algorithms, at NNLO and beyond.