In this study, a new and efficient numerical algorithm is developed to solve both the two-dimensional and three-dimensional compressible hydrodynamic stability problem. A parametric study of free shear flows with two or more supersonic streams is performed. Flows examined included shear layers, jets/wakes, and various geometrical combinations of these flows. The effect of Mach number on the stability characteristics of the flow is studied and found to confirm the work of other researchers who found that increasing the relative (or convective) Mach number increases the stability of the flow. For 2-D mean flows, the most amplified disturbance is shown to be axial for M<1.2 and fully three-dimensional for M> 1.2. Disturbances for three-dimensional mean flows are found here to be axial in the presence of side walls. The variation of the eigenfunctions and flow field disturbances as a function of Mach number and the flow geometry was also studied. Comparisons of the stability code results are also made to several turbulent mixing experiments. The stability code correctly predicts which parameters will accelerate mixing. New correlations of the effects of some important parameters on stability are developed. / Ph. D.