Diffusion models have emerged as a powerful family of deep generative models with record-breaking performance in various applications, including image synthesis, video generation, and molecule design. This survey provides an overview of the rapidly expanding body of work on diffusion models, categorizing research into three key areas: efficient sampling, improved likelihood estimation, and handling data with special structures. It also discusses the potential for combining diffusion models with other generative models for enhanced results and reviews their wide-ranging applications in fields such as computer vision, natural language processing, and interdisciplinary applications. The survey aims to provide a contextualized, in-depth look at the state of diffusion models, identifying key areas of focus and pointing to potential areas for further exploration. The paper covers the foundations of diffusion models, including denoising diffusion probabilistic models (DDPMs), score-based generative models (SGMs), and stochastic differential equations (Score SDEs). It then discusses efficient sampling methods, improved likelihood estimation, and handling data with special structures. The survey also examines connections with other generative models and reviews applications in computer vision, natural language generation, multi-modal generation, temporal data modeling, robust learning, and interdisciplinary applications. The paper concludes with future directions for research in diffusion models.Diffusion models have emerged as a powerful family of deep generative models with record-breaking performance in various applications, including image synthesis, video generation, and molecule design. This survey provides an overview of the rapidly expanding body of work on diffusion models, categorizing research into three key areas: efficient sampling, improved likelihood estimation, and handling data with special structures. It also discusses the potential for combining diffusion models with other generative models for enhanced results and reviews their wide-ranging applications in fields such as computer vision, natural language processing, and interdisciplinary applications. The survey aims to provide a contextualized, in-depth look at the state of diffusion models, identifying key areas of focus and pointing to potential areas for further exploration. The paper covers the foundations of diffusion models, including denoising diffusion probabilistic models (DDPMs), score-based generative models (SGMs), and stochastic differential equations (Score SDEs). It then discusses efficient sampling methods, improved likelihood estimation, and handling data with special structures. The survey also examines connections with other generative models and reviews applications in computer vision, natural language generation, multi-modal generation, temporal data modeling, robust learning, and interdisciplinary applications. The paper concludes with future directions for research in diffusion models.