Explore the complex world of diffeomorphism groups of discs in this André Aisenstadt Prize Lecture delivered by Alexander Kupers from the University of Toronto. Discover how these seemingly simple manifolds possess surprisingly intricate groups of diffeomorphisms through an examination of geometric, topological, and dynamical techniques used to understand these groups in low dimensions. Learn about the fascinating connections between these groups and stable homotopy theory and number theory in high dimensions, while gaining insight into recent breakthrough discoveries regarding their rational homotopy type. Benefit from a presentation designed for a broad mathematical audience that bridges multiple areas of mathematics including differential geometry, algebraic topology, and mathematical dynamics.