Obstructions to the existence of M{\o}ller maps

M{\o}ller maps are identifications between the observables of a perturbatively interacting physical system and the observables of its underlying free (i.e. non-interacting) system. This work studies and characterizes obstructions to the existence of such identifications. The main results are existence and importantly also non-existence theorems, which in particular imply that M{\o}ller maps do not exist for non-Abelian Chern-Simons and Yang-Mills theories on globally hyperbolic Lorentzian manifolds. These results are obtained through homological algebra techniques which are of independent interest in the analysis of classical field theories.