Projective structures on curves appear naturally in many areas of mathematics, from extrinsic conformal geometry to analysis, where the main problem is to find qualitative information about the solutions of Hill equations. In this paper, we describe in detail the correspondence between different equivalent definitions of projective structures and their isomorphism classes, correcting long-standing inexactitudes in the literature. As an application, we show that the Yamabe problem for curves in a conformal/Möbius ambient space has no solutions in general.