Peer review is considered an essential, if not defining aspect of science. However, ever since its inception, the system has been under constant critique and development. Also today, diverse efforts are underway to reform the journal peer review system. In combination with a growing interest in open science practices, open peer review (OPR) has become of central concern to multiple stakeholders within the scholarly communication process. However, what open peer review is understood to encompass and how effective some of its elements are in meeting the expectations of the peer review system, is uncertain. In this talk, I will give a brief overview of the history of peer review as a system, building up to the state where openness is considered to be the solution to some of its problems. I will then discuss a recently conducted systematic review to summarise the state of affairs of research on OPR to date. Last, I will introduce a new project studying the potential implications of open science practices, including OPR, for public trust in science.