Diverse efforts are underway to reform the journal peer review system. Combined with growing interest in Open Science practices, Open Peer Review (OPR) has become of central concern to the scholarly community. However, what OPR is understood to encompass and how effective some of its elements are in meeting the expectations of diverse communities, are uncertain. This scoping review updates previous efforts to summarize research on OPR to May 2022. Following the PRISMA methodological framework, it addresses the question: "What evidence has been reported in the scientific literature from 2017 to May 2022 regarding uptake, attitudes, and efficacy of two key aspects of OPR (Open Identities and Open Reports)?" The review identifies, analyses and synthesizes 52 studies matching inclusion criteria, finding that OPR is growing, but still far from common practice. Our findings indicate positive attitudes towards Open Reports and more sceptical approaches to Open Identities. Changes in reviewer behaviour seem limited and no evidence for lower acceptance rates of review invitations or slower turnaround times is reported in those studies examining those issues. Concerns about power dynamics and potential backfiring on critical reviews are in need of further experimentation. We conclude with an overview of evidence gaps and suggestions for future research. Also, we discuss implications for policy and practice, both in the scholarly communications community and the research evaluation community more broadly.