Geometric deep learning (GDL) has gained significant attention in scientific fields, for its proficiency in modeling data with intricate geometric structures. However, very few works have delved into its capability of tackling the distribution shift problem, a prevalent challenge in many applications. To bridge this gap, we propose GeSS, a comprehensive benchmark designed for evaluating the performance of GDL models in scientific scenarios with distribution shifts. Our evaluation datasets cover diverse scientific domains from particle physics, materials science to biochemistry, and encapsulate a broad spectrum of distribution shifts including conditional, covariate, and concept shifts. Furthermore, we study three levels of information access from the out-of-distribution (OOD) test data, including no OOD information, only unlabeled OOD data, and OOD data with a few labels. Overall, our benchmark results in 30 different experiment settings, and evaluates 3 GDL backbones and 11 learning algorithms in each setting. A thorough analysis of the evaluation results is provided, poised to illuminate insights for GDL researchers and domain practitioners who are to use GDL in their applications.