|Nowadays, the prevailing trend in the science-society relationship is to engage with the broader public, which is beneficial for the public, scientific institutes, scientific findings, and the legitimacy of science as a whole. This article provides a broad review of the rapidly growing research on Open Science and identifies the gaps in the current knowledge for future research. The review focuses on the science-society relationship, such that knowledge from this field is summarised and systematised. Insight into the most salient topics, including science communication, public engagement with science, public cognition of science, and challenges and potential unintended consequences connected to interactions with the public are examined. The first section of the paper focuses on science communication which involves efforts and approaches to inform the public about science by the most effective means. The section on public engagement reviews how scientists and scientific institutions are increasingly involved in direct interactions with the public and different groups of stakeholders to make science more open. The section focusing on public cognition of science provides information about public knowledge, perception, and trust regarding science, which both determines and is formed by public engagement. Last, risks, ethical issues, and data issues connected to the implementation of Open Science principles are reviewed, as there are many unintended consequences of Open Science which are examined by this current research. In conclusion, research covering the science-society relationship is rapidly growing. However, it brings multiple challenges as well as opportunities which are captured and discussed in a variety of existing studies. This article provides a coherent overview of this field in order to bring more comprehensible knowledge to scientists, scientific institutions, and outreach professionals.