As a major powerhorse for generative AI, diffusion models have demonstrated great successes in Euclidean spaces, such as for generating images and videos. This talk, on the other hand, will focus on a more nascent aspect, namely non-Euclidean diffusion models. One can for example consider the generative modeling of data that are discrete, living on manifold, constrained, or with multiple such modalities, as they correspond to important applications, some just emerging, including (vision-) language model, robotic motion planning, molecular engineering, and the design of quantum systems. After briefly introducing selected works of ours on these topics, I will expand on one example, where data live on a special type of manifolds known as Lie groups. Such a setting arises in the Gen-AI design of protein, robotic planning, and quantum problems. By leveraging and meshing variational optimization, delicate interplays between continuous- and discrete-time dynamics, and deep connections between optimization, sampling and optimal transport, I will turn our recent accelerated manifold optimization technique, first into a sampler that is fast converging without requiring log-concavity condition or its common relaxations, and then into an efficacious Lie group generative model. If time permits, theoretical understandings of selected diffusion models will also be briefly discussed.