Mathematical theory and computer codes have played a significant role in reducing wave drag and shock losses by reshaping the fuselage, wings, nacelles, and pylons as well as the airfoils appearing in propellers, compressors, and turbines. This chapter review major aspects of a code for the design of shockless airfoils in two-dimensional transonic flow. The code is based on a method of analytic continuation into the domain of two complex characteristic variables that enables one to associate a shockless airfoil with a given pressure distribution. The most interesting application of the code for calculation of shockless airfoils in cascade is to the design of supercritical compressor blades. Wind tunnel tests have already been performed successfully on an early example of such a cascade. The experimental data show great promise for this concept as a procedure to reduce losses in turbomachinery.