A computational investigation was undertaken to examine techniques for predicting supersonic dynamic flows, involving unsteadiness over fixed and moving surfaces. The fixed geometries examined were cylinder-flares and compression ramps, and the moving body geometries a pitching aerofoil and a rapidly deployed flap. Investigation into the characteristics of incipient separation of a supersonic cylinder-flare flow revealed that the separated length varied with a power of the flare angle and that the variation in height of the separated region varies in a bi-modal manner with flare angle. For small-scale separations (flare angles less than those which would traditionally have been expected to induce separation) the height of the separated region was seen to vary slowly with flare angle. For larger flare angles, the separation bubble was found to grow rapidly in height and length with increasing flare angle and produce significant deflection of the external flow. Computations of a Mach 5, compression ramp induced unsteady shock boundary layer interaction exhibited self-sustained oscillations at frequencies and amplitudes consistent with experimental data. Large dynamic structures (up to 1.7 boundary layer thicknesses in extent) were observed, and their production, propagation and deformation illustrated. By modifying the turbulent viscosities produced by a non-dimensional implementation of the Baldwin-Lomax turbulence model (using under- relaxation) a turbulence model was produced which accurately predicted separation lengths for a series of Mach 6.85 compression ramp flows encompassing laminar, transitional and turbulent flow regimes (dependent on ramp angle). A technique was developed to enable efficient computation of dynamically moving and/or deforming body flows. This technique was based on hierarchical, adaptive mesh refinement coupled with automatic generation of body surfaces, in which mesh adaption was used to capture the body geometry to within a specified accuracy. This, in conjunction with automatic cell creation and destruction, enabled the derivation of steady and unsteady, time accurate, conservative boundary conditions. This algorithm was used to compute a quasi-steady laminar supersonic pitching aerofoil flow, and an unsteady turbulent supersonic flap deployment. In both cases agreement with experiment was found to be good.