The main part of this dissertation is concerned with the development of a fully three dimensional compressible inverse design method, suitable for the design of radial-inflow turbines and other turbomachines with arbitrary meridional geometry. The basic idea of the method is to represent the action of the blades by sheets of vorticity whose strength is determined from prescribed distribution of rVθ (or circulation 2πrVθ). The flow is assumed subsonic and inviscid and the blades are assumed to have negligible thickness. But the blade blockage effects are approximately accounted for by using a mean streamsurface thickness parameter in the continuity equation. As a first approach, the pitchwise variation of density was neglected and the problem was solved by using an approximate form of the continuity equation. Simple expressions were derived for the terms neglected in the approximate continuity equation. The problem was also solved by using the exact form of the continuity equation and the results of the approximate and exact methods were compared for a number of test cases. The comparison showed that the approximate method can compute the blade shape accurately. A small high (subsonic) speed radial-inflow turbine was designed by the new method. In order to assess the accuracy of the method, the flow through the designed impeller was computed by three dimensional inviscid and viscous flow analysis programs and good correlation was obtained between the computed and specified rVθ distributions. The designed impeller was manufactured and its performance was measured and compared to three other baseline impellers, one conventional and two experimental. The new impeller performed substantially better than all the baseline turbines and showed a 5.5% improvement over the conventional impeller. However, only 2.5% of this improvement was attributed to the aerodynamically superior blade shape designed by the new method. An appreciable improvement in efficiency was also observed at off-design conditions. Finally, the presence of a, generally observed, high loss region near the shroud at exit of radial-inflow turbines was investigated and it was found that secondary flow is the basic mechanism behind this phenomenon.