Generative AI is a rapidly evolving technology that has garnered significant interest lately. In this presentation, we'll discuss the latest approaches, organizing them within a cohesive framework using stochastic differential equations to understand complex, high-dimensional data distributions. We'll highlight the necessity of studying generative models beyond Euclidean spaces, considering smooth manifolds essential in areas like robotics and medical imagery, and for leveraging symmetries in the de novo design of molecular structures. Our team's recent advancements in this blossoming field, ripe with opportunities for academic and industrial collaborations, will also be showcased.