How does the structure of the peer review process, which can vary among journals, influence the quality of papers published in a journal? This article studies multiple systems of peer review using computational simulation. I find that, under any of the systems I study, a majority of accepted papers are evaluated by an average reader as not meeting the standards of the journal. Moreover, all systems allow random chance to play a strong role in the acceptance decision. Heterogeneous reviewer and reader standards for scientific quality drive both results. A peer review system with an active editor—that is, one who uses desk rejection before review and does not rely strictly on reviewer votes to make decisions—can mitigate some of these effects.