Diffusion models have emerged as a powerful family of deep generative models, achieving state-of-the-art performance in various applications such as image synthesis, video generation, and molecule design. This survey provides an overview of the rapidly expanding research on diffusion models, categorizing it into three key areas: efficient sampling, improved likelihood estimation, and handling data with special structures. The survey also discusses the potential for combining diffusion models with other generative models to enhance results. Additionally, it reviews the wide-ranging applications of diffusion models in fields such as computer vision, natural language processing, temporal data modeling, and interdisciplinary applications. The survey aims to provide a comprehensive and in-depth look at the state of diffusion model research, identifying key areas of focus and potential areas for future exploration.Diffusion models have emerged as a powerful family of deep generative models, achieving state-of-the-art performance in various applications such as image synthesis, video generation, and molecule design. This survey provides an overview of the rapidly expanding research on diffusion models, categorizing it into three key areas: efficient sampling, improved likelihood estimation, and handling data with special structures. The survey also discusses the potential for combining diffusion models with other generative models to enhance results. Additionally, it reviews the wide-ranging applications of diffusion models in fields such as computer vision, natural language processing, temporal data modeling, and interdisciplinary applications. The survey aims to provide a comprehensive and in-depth look at the state of diffusion model research, identifying key areas of focus and potential areas for future exploration.