New and powerful methods of characterizing existing and new airfoil geometries with mathematical equations are presented. The methods are applicable to a wide range of airfoil shapes representing traditional, cusped, reflexed, flat-bottom, laminar, transonic, and supersonic designs. With the emphasis on low-speed airfoils, several existing airfoils are first closely matched with the math-modeling methods. Then, to support the design of new airfoil geometries, a new interpretation of Theodorsen's potential flow method is outlined for the calculation and presentation of surface velocity in inviscid flow. Also, a vector approach is introduced for the calculation of pitching moment. Finally, new math-modeled airfoils are proposed for conventional and unique aircraft configurations.