A fully elliptic computational method for the analysis of steady viscous flow in high speed subsonic centrifugal compressor impellers with tip leakage, is presented. A generalised curvilinear, non-orthogonal grid is utilised and the timeaveraged Navier-Stokes equations are transformed and expressed in a fully conservative form. The discretisation of the governing equations is performed through finite volume integration. The solution procedure employs a non-staggered variable arrangement and a SIMPLE based method for coupling the velocity and pressure fields. The turbulence effects are simulated with the use of the k-e model, modified to account for rotation and streamline curvature, and the near-wall viscous phenomena are modelled through the wall function method. The numerical model is implemented for the flow prediction in a series of two and three dimensional test cases. Incompressible flow predictions in twodimensional cascades and three-dimensional ducting systems with different geometrical features and inlet conditions are initially performed and the numerical results are compared against available experimental data. The final objective of the present study is achieved through the comparative study of the predictions obtained against the results of Eckardt's experimental investigation of the viscous compressible flow in a high speed radial impeller operating at design condition and in a backswept impeller at design and off-design conditions. In addition, the flow is simulated in the passages of the Rolls Royce GEM impeller which was tested at Cranfield at design and off-design flow rates. A jet/wake pattern was discerned in all the simulated centrifugal compressor cases and a good overall agreement was achieved with the measured wake formation and development; and, encouraging results were obtained on the evolution of the secondary flows. The tip leakage effects influenced the loss distribution, the size and the location of the wake flow pattern at the rotor exit. The effects of the flow mass rate on the detailed flow pattern and on the compressor performance have been well represented. In certain cases, the quality of the present predictions is an improvement over that obtained by other 'state-of-the-art' Navier-Stokes solvers. In conclusion, the developed finite volume flow model has captured a large number of complex flow phenomena encountered in the tested impellers and is expected to provide a useful aerodynamic analysis tool for stationary or rotating, axial or radial turbomachinery components.