In 1968, D. Atkinson proved in a series of papers the existence of functions satisfying all known constraints of the S-matrix bootstrap for the 2-to-2 S-matrix of gapped theories. To date, this is the only result of this sort, while a contrario no current technology allows to generate, even numerically, fully consistent S-matrices in d>2. Beyond the mathematical results themselves, the proof, based on establishing the existence of a fixed point of a certain map, also suggests a procedure to be implemented numerically and which would produce fully consistent S-matrix functions via iterating dispersion relations, and using as an input a quantity related to the inelasticity of a given scattering process. In this talk, I will report on some work being finalised, done in collaboration with A. Zhiboedov, about analytical and numerical aspects of developing and implementing this scheme. I will review basic concepts of the S-matrix program and show some of our results on non-perturbative scalar, phi^4-like S-matrices in 4, describe their properties and compare to other approaches in the literature. If time allows, I will present some results in 3 dimensions and discuss subtle aspects of the high energy (Regge behaviour) of the S-matrices.