Open Science (OS) aims, in part, to drive greater societal impact of academic research. Government, funder and institutional policies state that it should further democratize research and increase learning and awareness, evidence-based policy-making, the relevance of research to society's problems, and public trust in research. Yet, measuring the societal impact of OS has proven challenging and synthesized evidence of it is lacking. This study fills this gap by systematically scoping the existing evidence of societal impact driven by OS and its various aspects, including Citizen Science (CS), Open Access (OA), Open/FAIR Data (OFD), Open Code/Software and others. Using the PRISMA Extension for Scoping Reviews and searches conducted in Web of Science, Scopus and relevant grey literature, we identified 196 studies that contain evidence of societal impact. The majority concern CS, with some focused on OA, and only a few addressing other aspects. Key areas of impact found are education and awareness, climate and environment, and social engagement. We found no literature documenting evidence of the societal impact of OFD and limited evidence of societal impact in terms of policy, health, and trust in academic research. Our findings demonstrate a critical need for additional evidence and suggest practical and policy implications.