A practical procedure for the optimum design of low-speed airfoils is demonstrated. The procedure uses an optimization program based on a gradient algorithm coupled with an aerodynamic analysis program that uses a relaxation solution of the in viscid, full-potential equation. The analysis program is valid for both incompressible and compressible flow, thereby making optimum design of high-speed, shock-free airfoils possible. Results are presented for the following three constrained optimization problems at fixed angle of attack and Mach number: (i) adverse pressure-gradient minimization, (ii) pitching-moment minimization, and (iii) lift maximization. All three optimization problems were studied with various aerodynamic and geometric constraints.