The post-Newtonian expansions are one of the most important techniques in general relativity for extracting physical predictions. As these expansions are based on formal calculations, it is of fundamental importance to understand when and in what sense they yield valid approximations. I propose to develop a rigorous approach to the post-Newtonian expansions. A successful outcome would supply a mathematical foundation for the expansions and also identify the situations where the expansions fail. Determining when the post-Newtonian expansions fail is important as it identifies new physical situations that will have been left unaccounted for by the formal calculations.