Compression by shock waves is a specific way of compressing gases. It has been practically applied for many years in supersonic flying objects. The idea of using this method in rotary engines is extremely appealing because one disk can replace several or a dozen disks of an axial compressor, significantly reducing the weight and production costs of the engine and lowering the fuel consumption due to possible increased compression ratio. This paper presents a review of existing technical solutions and the results of published research devoted to the construction of shock wave compression rotary engines: patents, scientific publications describing various research methods, numerical calculations, and the experimental results of unusual technical solutions. The characteristic solutions and problems that arose during the implementation of these methods are presented and described. Judging from the presented overview, these have wide application possibilities, and an enormous intellectual and financial effort has been put into the construction of such engines. Conversely, there is a rather hermetic group of scientists involved in this activity.