Abstract: This talk explores the interplay between model-based guarantees and learning-based flexibility in the control of dynamical systems. I begin with safety-critical control using control barrier functions (CBFs), highlighting that while CBFs enforce state constraints, they may induce unstable internal dynamics. I introduce conditions under which CBF-based safety filters ensure boundedness of the full system state. I then transition to learning representations of hybrid dynamical systems. I present a framework that learns continuous neural representations by exploiting the geometric structure induced by guards and resets, enabling accurate flow prediction without explicit mode switching. Finally, I discuss generative learning approaches for control, emphasizing guided diffusion models that jointly represent states and actions. Through applications to agile humanoid locomotion, motion synthesis, and dynamic manipulation, I demonstrate how generative models can produce versatile, long-horizon behaviors while respecting physical constraints. Together, these results highlight how structure, geometry, and learning can bridge safety guarantees and expressive control in complex dynamical systems.