Open peer review (OPR) is a cornerstone of the emergent Open Science agenda. Yet to date no large-scale survey of attitudes towards OPR amongst academic editors, authors, reviewers and publishers has been undertaken. This paper presents the findings of an online survey, conducted for the OpenAIRE2020 project during September and October 2016 that sought to bridge this information gap in order to aid the development of appropriate OPR approaches by providing evidence about attitudes towards and levels of experience with OPR. The results of this cross-disciplinary survey, which received 3,062 full responses, show the majority of respondents to be in favour of OPR becoming mainstream scholarly practice, as they also are for other areas of Open Science, like Open Access and Open Data. We also observe surprisingly high levels of experience with OPR, with three out of four (76.2%) respondents reporting having taken part in an OPR process as author, reviewer or editor. There were also high levels of support for most of the traits of OPR, particularly open interaction, open reports and final-version commenting. Respondents were against opening reviewer identities to authors, however, with more than half believing it would make peer review worse. Overall satisfaction with the peer review system used by scholarly journals seems to strongly vary across disciplines. Taken together, these findings are very encouraging for OPR’s prospects for moving mainstream but indicate that due care must be taken to avoid a “one-size fits all” solution and to tailor such systems to differing (especially disciplinary) contexts. More research is also needed. OPR is an evolving phenomenon and hence future studies are to be encouraged, especially to further explore differences between disciplines and monitor the evolution of attitudes. Note: This report is a pre-print of an article intended for peer-reviewed publication. The authors gratefully invite comments until 22 May 2017 on the OpenAIRE blog (https://blogs.openaire.eu/?p=1895) - please leave any reviews/comments in the comments section at the foot of the page. All feedback will be gratefully received.