Sub-Riemannian geometry studies spaces where motion is restricted to certain directions, generalizing Riemannian geometry. It grew out of 19th-century work by Carathéodory and later developments by Chow, Rashevsky, and Hörmander, with links to control theory and analysis.

This talk will briefly review the main ideas and classical results, including the Chow–Rashevsky theorem and basic examples arising from group theory, classical mechanics, and mathematical physics. I will then discuss current topics related to the regularity and uniqueness of geodesics, including the role and structure of abnormal minimizers, the definition and understanding of curvature in the absence of a full Riemannian metric, and questions in metric geometry.