Diffusion models, a powerful generative AI technology, have achieved significant success in various domains such as computer vision, audio, reinforcement learning, and computational biology. Despite their empirical success, the theoretical foundation of diffusion models is limited, which hinders principled methodological innovations. This paper reviews the emerging applications of diffusion models, their statistical properties, and sampling capabilities. It also explores the theoretical progress on unconditional and conditional diffusion models, including score function learning, approximation, and estimation. The paper further discusses the use of diffusion models in high-dimensional structured optimization and black-box optimization. Finally, it outlines future directions, including connections to stochastic control, adversarial robustness, and discrete diffusion models. The goal is to provide a comprehensive theoretical exposure to stimulate forward-looking research on diffusion models.Diffusion models, a powerful generative AI technology, have achieved significant success in various domains such as computer vision, audio, reinforcement learning, and computational biology. Despite their empirical success, the theoretical foundation of diffusion models is limited, which hinders principled methodological innovations. This paper reviews the emerging applications of diffusion models, their statistical properties, and sampling capabilities. It also explores the theoretical progress on unconditional and conditional diffusion models, including score function learning, approximation, and estimation. The paper further discusses the use of diffusion models in high-dimensional structured optimization and black-box optimization. Finally, it outlines future directions, including connections to stochastic control, adversarial robustness, and discrete diffusion models. The goal is to provide a comprehensive theoretical exposure to stimulate forward-looking research on diffusion models.