When electromagnetic fields strike sharp surface discontinuities, such as edges and corners, they create a diffraction phenomenon. In high-frequency methods, the field generated when a wave impinges upon a PEC wedge can be accounted by using Geometrical Optics (GO), based more on Snell's Law of Reflection, and Diffraction, based on Fermat's Principle for diffraction . The diffraction phenomenon and its field contributions can be accounted for by utilizing diffraction coefficients for the appropriate discontinuities. One set of diffraction coefficients were derived by Keller , and they were dubbed as Keller's diffraction coefficients based on Geometrical Theory of Diffraction (GTD). These diffraction coefficients exhibited singularities along the incident and reflection shadow boundaries (ISB and RSB), and limited their use. Another set of diffraction coefficients were derived by Kouyoumjian and Pathak , and they were dubbed as the K & P diffraction coefficients based on the Uniform Theory of Diffraction (UTD). The UTD diffraction coefficients eliminated the singularities along the ISB and RSB and provided smooth transition from one side to the other side of the respective shadow boundaries. This was accomplished by introducing transition functions in the UTD formulation which removed the singularities and provided a more accurate representation of the field at and near the shadow boundaries. The UTD diffraction coefficients received international acclaim and expanded the application of diffraction theory to many otherwise intractable boundary-value problems, from simple geometries (such as strips and ground planes) to more complex (such as airframes, ships, and ground vehicles). Some of these applications will be illustrated in this presentation.