A method for solving general compressible three-dimensional boundary-layer flows is developed. The equations are initially placed in a Crocco-type form which makes use of similarity variables. A general solution technique which employs an implicit finite-difference scheme that is stable for negative transverse velocities is developed. The solutions for some 10° vertical half-angle elliptical cones of varying ellipticity ratios at up to 8° angle of attack at a Mach number of 7.95 are presented. Skin friction and heat-transfer-rate distributions are presented for all cases. Good agreement has been found with experiment for the heat-transfer rates on circular cones at angle of attack. It has also been determined that increasing ellipticity tended to produce large heat-transfer-rate gradients, and large increases in the peak heat-transfer rates and peak skin frictions. It has also been found that for even moderate angles of attack and small ellipticity ratios the peak heat-transfer rate and peak longitudinal skin friction fall close to the major axis of the cone. / Master of Science