In the design of the hypersonic airliner which can greatly shorten the flight time and conduct space travel, it is of great significance for a Riemann solver to accurately simulate hypersonic flows. In this survey, the research process on the Riemann solver for hypersonic flows is reviewed, including the constructions of the traditional Riemann solvers, improvements of the traditional Riemann solvers for the shock anomaly, and the importance of the all-speed Riemann solver for hypersonic flows. Moreover, constructions and applications of the all-speed Riemann solvers are presented. It is obvious that the Riemann solver should be capable of obtaining physical solutions at low speeds and be robust against shock anomaly near strong shocks at the same time in simulating hypersonic flows.
