A numerical method has been developed to solve the steady three dimensional Reynolds-averaged Navier-Stokes equations by successive approximations. The method is applicable to rotors of axial or radial compressors and turbines. Viscous flow fields were calculated in the blading passages of an axial supersonic compressor cascade, a radial impeller and a backswept impeller. Pressure measurements on the cascade blades agreed with corresponding computations. Radial impeller calculations showed suction blade separation, while backswept calculations indicated shroud separation. This paper describes the numerical method in detail, with results presented as numerical examples.