The public-key model for interactive proofs has proved to be quite effective in improving protocol efficiency [CGGM00]. We argue, however, that its soundness notion is more subtle and complex than in the classical model, and that it should be better understood to avoid designing erroneous protocols. Specifically, for the public-key model, we – identify four meaningful notions of soundness; – prove that, under minimal complexity assumptions, these four notions are distinct; – identify the exact soundness notions satisfied by prior interactive protocols; and – identify the round complexity of some of the new notions.