The purpose of this paper is to present the numerical results of several theoretical, aerodynamic investigations of impellers in radial and mixed-flow centrifugal compressors and to discuss the significance of these results with respect to improved compressor performance. The solutions are not intended to supply specific design information but rather to help create a general understanding and awareness of the flow conditions in impellers of centrifugal compressors. The paper is given in two parts. In Part 1 two-dimensional relaxation solutions for compressible and incompressible fluids are presented for flow with axial symmetry (two solutions) and for flow between blades on surfaces of revolution about the axis of the impeller (eight solutions). The effects of variations in the following design and operating conditions of the impeller are investigated: (a) Compressor flow rate, (b) impeller-tip speed, (c) number of blades, and (d) blade curvature on flow surfaces of revolution. The numerical results are presented in plots of the streamlines, constant-velocity or Mach number lines, and constant static-pressure lines. In Part 2 three approximate methods of solution are presented which can be used for relatively rapid analysis of compressible flow between blades on flow surfaces of revolution.