In recent years, the scientific community has called for improvements in the credibility, robustness and reproducibility of research, characterized by increased interest and promotion of open and transparent research practices. While progress has been positive, there is a lack of consideration about how this approach can be embedded into undergraduate and postgraduate research training. Specifically, a critical overview of the literature which investigates how integrating open and reproducible science may influence student outcomes is needed. In this paper, we provide the first critical review of literature surrounding the integration of open and reproducible scholarship into teaching and learning and its associated outcomes in students. Our review highlighted how embedding open and reproducible scholarship appears to be associated with (i) students' scientific literacies (i.e. students' understanding of open research, consumption of science and the development of transferable skills); (ii) student engagement (i.e. motivation and engagement with learning, collaboration and engagement in open research) and (iii) students' attitudes towards science (i.e. trust in science and confidence in research findings). However, our review also identified a need for more robust and rigorous methods within pedagogical research, including more interventional and experimental evaluations of teaching practice. We discuss implications for teaching and learning scholarship.