Solar geoengineering describes a set of proposals to deliberately alter the earth’s radiative balance to reduce climate risks. We elicit judgements on natural science research priorities for solar geoengineering through a survey and in-person discussion with 72 subject matter experts, including two thirds of all scientists with ≥10 publications on the topic. Experts prioritized Earth system response (33%) and impacts on society and ecosystems (27%) over the human and social dimensions (17%) and developing or improving solar geoengineering methods (15%), with most allocating no effort to weather control or counter-geoengineering. While almost all funding to date has focused on geophysical modeling and social sciences, our experts recommended substantial funding for observations (26%), perturbative field experiments (16%), laboratory research (11%) and engineering for deployment (11%). Of the specific proposals, stratospheric aerosols received the highest average priority (34%) then marine cloud brightening (17%) and cirrus cloud thinning (10%). The views of experts with ≥10 publications were generally consistent with experts with <10 publications, though when asked to choose the radiative forcing for their ideal climate scenario only 40% included solar geoengineering compared to 70% of experts with <10 publications. This suggests that those who have done more solar geoengineering research are less supportive of its use in climate policy. We summarize specific research recommendations and challenges that our experts identified, the most salient of which were fundamental uncertainties around key climate processes, novel challenges related to solar geoengineering as a design problem, and the challenges of public and policymaker engagement.