The method described here is a higher-order three-dimensional surface panel method. It utilizes four-sided panels on the actual surface of the body about which flow is to be computed and is thus applicable to arbitrary configurations. In contrast to a first-order panel method that uses flat panels, usually with constant source and/or vorticity density, the present method uses curved panels of second degree with linearly varying source and vorticity density. The mathematical problem solved is that of incompressible potential flow but the calculation can be extended to subsonic flow by means of well-known compressibility corrections. This report presents a discussion of the techniques adopted and some alternatives. A rigorous derivation of the higher-order formulation is carried out. All formulas and logic required by the method are included, except that the report on the corresponding first-order method is relied on for formulas and procedures common to the two techniques. First-order and higher-order calculated results are compared with each other and with exact solutions.