Peer review has never been a uniform practice, but is now more diverse than ever. Despite a vast literature, little is known of how different disciplines organise peer review. This paper draws on 95 qualitative interviews with editors and publishers and several hundred written reports to analyse the organisation of peer review in pure mathematics. This article focuses on the practice of ‘quick opinions’ at top journals in mathematics: asking (senior) experts about a paper’s importance, and only after positive evaluation sending the paper for a full review (which most importantly means checking the paper’s correctness). Quick opinions constitute a form of ‘importance only’ peer review and are thus the opposite of the ‘soundness only’ approach at mega-journals such as PLOS ONE. Quick opinions emerged in response to increasing submissions and the fact that checking correctness in mathematics is particularly time-consuming. Quick opinions are informal and are often only addressed to editors. They trade on, indeed reinforce, a journal hierarchy, where journal names are often used as a ‘members’ measurement system’ to characterise importance. Finally, quick opinions highlight that a key function of the peer-reviewed journal today, apart from validation and filtration, is ‘designation’ – giving authors items on their CV.