In this paper I consider two accounts of scientific discovery, Robert Hudson’s and Peter Achinstein’s. I assess their relative success and I show that while both approaches are similar in promising ways, and address experimental discoveries well, they could address the concerns of the discovery sceptic more explicitly than they do. I also explore the implications of their inability to address purely theoretical discoveries, such as those often made in mathematical physics. I do so by showing that extending Hudson’s or Achinstein’s account to such cases can sometimes provide a misleading analysis about who ought to be credited as a discoverer. In the final sections of the paper I work out some revisions to the Hudson/Achinstein account by drawing from a so-called structural realist view of theory change. Finally, I show how such a modified account of discovery can answer sceptical critics such as Musgrave or Woolgar without producing misleading analyses about who ought to receive credit as a discoverer in cases from the mathematical sciences. I illustrate the usefulness of this approach by providing an analysis of the case of the discovery of the Casimir effect.