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ABSTRACT Diffusion models, a powerful and universal generative artificial intelligence technology, have achieved tremendous success and opened up new possibilities in diverse applications. In these applications, diffusion models provide flexible high-dimensional data modeling, and act as a sampler for generating new samples under active control towards task-desired properties. Despite the significant empirical success, theoretical underpinnings of diffusion models are very limited, potentially slowing down principled methodological innovations for further harnessing and improving diffusion models. In this paper, we review emerging applications of diffusion models to highlight their sample generation capabilities under various control goals. At the same time, we dive into the unique working flow of diffusion models through the lens of stochastic processes. We identify theoretical challenges in analyzing diffusion models, owing to their complicated training procedure and interaction with the underlying data distribution. To address these challenges, we overview several promising advances, demonstrating diffusion models as an efficient distribution learner and a sampler. Furthermore, we introduce a new avenue in high-dimensional structured optimization through diffusion models, where searching for solutions is reformulated as a conditional sampling problem and solved by diffusion models. Lastly, we discuss future directions about diffusion models. The purpose of this paper is to provide a well-rounded exposure for stimulating forward-looking theories and methods of diffusion models.