[Significance] The supercritical carbon dioxide (S-CO2) Brayton cycle is a power cycle at intermediate temperature and high pressure. This cycle is considered an important solution to improving the efficiency of traffic power systems such as gas turbines and internal combustion engines by recycling exhaust energy at high temperatures. The compressor is considered one of the most important components of this cycle. Its efficient and stable operation plays an important role in cycle performance. [Progress] In this paper, the research progress on S-CO2 centrifugal compressor flow characteristics was reviewed from four aspects: experiment, one-dimensional flow analysis, three-dimensional flow characteristics, and flow control. Researchers learned from the experimental studies of the S-CO2 centrifugal compressor that the special thermophysical properties of S-CO2, particularly their dramatic change near the critical point, brought great challenges to the design and stable operation of this compressor. Therefore, the problems of compressor flow caused by the drastic physical properties change near the critical point of the working medium, and the related research contents were emphatically expounded. The current research on one-dimensional flow analysis of the S-CO2 centrifugal compressor is mainly conducted by the one-dimensional mean streamline method considering the special thermophysical properties of S-CO2 fluid. The preliminary aerodynamic design of the S-CO2 centrifugal compressor was conducted using one-dimensional flow analysis. This method is limited by its prediction accuracy under off-design conditions. In addition, the flow details inside the compressor could not be obtained by this method. To reveal the flow mechanism of the S-CO2 centrifugal compressor, its three-dimensional flow characteristics must be deeply understood, and its internal flow field information must be obtained. The research on the three-dimensional flow characteristics of the S-CO2 centrifugal compressor was mostly conducted by the computational fluid dynamics (CFD) numerical simulation method, which can be used to obtain the flow field of the centrifugal compressor and present the relevant flow phenomenon. Because of the drastic variations in the thermophysical properties of S-CO2 fluid near the critical point, special consideration was taken in the process of the CFD simulation of the flow inside the centrifugal compressor. By applying CFD to S-CO2 centrifugal compressor three-dimensional flow characteristics, researchers found that this special thermal physical property also brought special flow phenomena inside the flow domain of the S-CO2 centrifugal compressor. The research on S-CO2 centrifugal compressor three-dimensional flow characteristics mainly focused on its steady flow and needs to further reveal deeply and comprehensively the flow mechanism of the S-CO2 centrifugal compressor under various unsteady working conditions. The flow control of the S-CO2 centrifugal compressor was mainly by the passive flow control method using the relevant control method of the air compressor for reference, and its effect was remarkable. As for the active flow control method, few studies have heeded its effect on the S-CO2 centrifugal compressor. [Conclusions and Prospects] In this paper, the flow characteristics of the S-CO2 centrifugal compressor are summarized, and their research prospects are proposed. These flow characteristics are considerably different from those of centrifugal compressors with conventional fluids, mainly because of the special physical properties of S-CO2 fluid. In the future, more advanced research methods are expected to be used, such as visual flow experiments, one-dimensional flow analysis incorporating machine learning algorithms, and active flow control, to conduct more in-depth and comprehensive studies of S-CO2 compressor flow characteristics. © 2023 Press of Tsinghua University. All rights reserved.