Stochastic configuration network (SCN), as an emerging incremental neural network model, is different from other randomized neural network methods. It can configure the parameters of hidden layer nodes through supervision mechanisms, thereby ensuring the fast convergence performance of SCN. Due to the advantages of high learning efficiency, low human intervention, and strong generalization ability, SCN has attracted a large number of national and international scholars and developed rapidly since it was proposed in 2017. In this study, SCN research is summarized from the aspects of basic theories, typical algorithm variants, application fields, and future research directions of SCN. Firstly, the algorithm principles, universal approximation capacity, and advantages of SCN are analyzed theoretically. Secondly, typical variants of SCN are studied, such as DeepSCN, 2DSCN, Robust SCN, Ensemble SCN, Distributed SCN, Parallel SCN, and Regularized SCN. Then, the applications of SCN in different fields, including hardware implementation, computer vision, medical data analysis, fault detection and diagnosis, and system modeling and prediction are introduced. Finally, the development potential of SCN in convolutional neural network architectures, semi-supervised learning, unsupervised learning, multi-view learning, fuzzy neural network, and recurrent neural network is pointed out.