With the renewed interest in the supersonic and hypersonic flight vehicles, new inverse Euler methods are developed in these flow regimes where a space marching numerical technique is valid. In order to get a general understanding for the specification of target pressure distributions, a study of minimum drag body shapes was conducted over a Mach number range from 3 to 12. Numerical results show that the power law bodies result in low drag shapes, where the n=.69 (l/d = 3) or n=.70 (l/d = 5) shapes have lower drag than the previous theoretical results (n=.75 or n=.66 depending on the particular form of the theory). To validate the results, a numerical analysis was made including viscous effects and the effect of gas model. From a detailed numerical examination for the nose regions of the minimum drag bodies, aerodynamic bluntness and sharpness are newly defined. Numerous surface pressure-body geometry rules are examined to obtain an inverse procedure which is robust, yet demonstrates fast convergence. Each rule is analyzed and examined numerically within the inverse calculation routine for supersonic (M_∞= 3) and hypersonic (M_∞ = 6.28) speeds. Based on this analysis, an inverse method for fully three dimensional supersonic and hypersonic bodies is developed using the Euler equations. The method is designed to be easily incorporated into existing analysis codes, and provides the aerodynamic designer with a powerful tool for design of aerodynamic shapes of arbitrary cross section. These shapes can correspond to either "wing like" pressure distributions or to "body like" pressure distributions. Examples are presented illustrating the method for a non-axisymmetric fuselage type pressure distribution and a cambered wing type application. The method performs equally well for both nonlifting and lifting cases. For the three dimensional inverse procedure, the inverse solution existence and uniqueness problem are discussed. Sample calculations demonstrating this problem are also presented. / Ph. D.