Solar geoengineering is an emerging topic in climate-change discussions. To support future decisions on the deployment of this technology, society requires better estimates of its environmental impacts and limitations. As solar geoengineering has never been deployed, conclusions about its climatic effects are primarily obtained through models and natural analogues. As such, our confidence in projections of solar geoengineering, the basis for that confidence and how our confidence can be improved is limited. In this Perspective, we review our current understanding of uncertainty and risk in solar geoengineering via stratospheric aerosols. Using a risk-register framework, we illustrate key uncertainties, such as sub-grid-scale mixing or effects of stratospheric heating, investigations of which should be prioritized to transition the field to a mission-driven research agenda. We conclude with recommendations for possible avenues of research, including targeted model intercomparisons and appropriately governed small-scale field experiments.