Diffusion models are a class of deep generative models that have shown impressive results on various tasks with a solid theoretical foundation. Despite demonstrated success than state-of-the-art approaches, diffusion models often entail costly sampling procedures and sub-optimal likelihood estimation. Significant efforts have been made to improve the performance of diffusion models in various aspects. In this article, we present a comprehensive review of existing variants of diffusion models. Specifically, we provide the taxonomy of diffusion models and categorize them into three types: sampling-acceleration enhancement, likelihood-maximization enhancement, and data-generalization enhancement. We also introduce the other generative models (i.e., variational autoencoders, generative adversarial networks, normalizing flow, autoregressive models, and energy-based models) and discuss the connections between diffusion models and these generative models. Then we review the applications of diffusion models, including computer vision, natural language processing, waveform signal processing, multi-modal modeling, molecular graph generation, time series modeling, and adversarial purification. Furthermore, we propose new perspectives pertaining to the development of generative models. Github: this https URL.