In recent years, the appearance of the broad learning system (BLS) is poised to revolutionize conventional artificial intelligence methods. It represents a step toward building more efficient and effective machine-learning methods that can be extended to a broader range of necessary research fields. In this survey, we provide a comprehensive overview of the BLS in data mining and neural networks for the first time, focusing on summarizing various BLS methods from the aspects of its algorithms, theories, applications, and future open research questions. First, we introduce the basic pattern of BLS manifestation, the universal approximation capability, and essence from the theoretical perspective. Furthermore, we focus on BLS's various improvements based on the current state of the theoretical research, which further improves its flexibility, stability, and accuracy under general or specific conditions, including classification, regression, semisupervised, and unsupervised tasks. Due to its remarkable efficiency, impressive generalization performance, and easy extendibility, BLS has been applied in different domains. Next, we illustrate BLS's practical advances, such as computer vision, biomedical engineering, control, and natural language processing. Finally, the future open research problems and promising directions for BLSs are pointed out.