Instability waves in a free shear layer formed by two parallei compressibie streams are analyzed using the linear spatial stability theory. Both viscous and inviscid disturbances are considered. The basic state is obtained by solving the compressibie laminar boundary-layer equations or is specified by the hyperbolic tangent velocity profile. The effects of viscosity, Mach number, the velocity and temperature ratios on the growth rate are determined. Unlike the boundary layer flow, viscosity has a stabilizing effect on the mixing layer flow. Increasing the temperature ratio produces a strong stabilizing effect on the growth of the mixing flow; this stabilization does not, however, persist at higher Mach numbers. Whereas the maximum growth rate of the Incompressible mixing layer varies linearly with the velocity ratio, the maximum growth rate of the compressible mixing flow varies nonlinearly with the velocity ratio. The numerical results substantiate the fact that the convective Mach number Is the appropriate parameter for correlating the compressibility effects on the spreading rate of the mixing layer. The ratio of the spreading rate of a compressible layer to that of an incompressible layer at the same velocity and density ratios depends primarily on the convective Mach number. Three-dimensional waves become important when the convective Mach number is greater than 0.6. The influence of nonparallelism on the spatial growth rate of two-dimensional disturbances is evaluated and is found to be negligible. Linear subharmonic Instabilities of a compressible mixing layer, which Is spatially periodic in a translating frame of reference, are analyzed by using Floquet theory. The basic state is obtained by the linear superposition ofa steady mean flow, which is given by a solution to the compressible boundary-layer equations or by a hyperbolic tangent velocity profile approximation, and the neutral primary wave of that mean flow. The results show that the growth rates of two-dimensional subharmonic instabilities (pairing mode) increase with increasing amplitude of the periodicity but decrease with increasing the convective iVIach number. In the incompressible flow case, the most amplified subharmonic wave is a two-dimensional mode, which is in agreement with the published results. For subsonic convective Mach numbers, the presence of the periodicity enhances the growth rates of three-dimensional subharmonic waves over a wide range of spanwise wavenumber which shows a preferred band over which the growth rate is maximum. However, when the convective IVIach number is greater than one, the interaction between the subharmonic wave and the primary wave marginally increases the maximum growth rate of the subharmonic. Nevertheless, that interaction dramatically increases the range of amplified spanwise wave numbers. Fourth-order compact finite-difference codes are developed for solving the compressible boundary-layer equations and investigating their primary and subharmonic instabilities. The codes proved to be very accurate and versatile.