Abstract: One-relator groups G=F/<> pose a challenge to geometric group theorists. On the one hand, they satisfy strong algebraic constraints. On the other hand, they are not susceptible to geometric techniques, since some of them ? such as the famous Baumslag?Solitar groups ? exhibit extremely pathological behaviour. I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator w introduced by Puder ? the *primitivity rank* \pi(w). A sample application is that every subgroup of G of rank < \pi(w) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups. This is joint work with Larsen Louder.