Keller's geometrical theory of diffraction (GTD) represents a major breakthrough in solving a wide variety of electromagnetic (EM) radiation and scattering problems at high frequencies. In particular, the GTD is an extension of geometrical optics to include the class of diffracted rays via a generalization of Fermat's principle. These diffracted rays are initiated at certain localized parts of an illuminated surface, e.g., edges, vertices, and shadow boundaries associated with smooth convex bodies. However, being a purely ray optical theory, the original GTD fails within the transition regions adjacent to geometric optical shadow boundaries. This and other limitations of the GTD can be overcome via a new uniform version of the GTD-or the UTD, which requires the diffracted field to make the total high frequency field continuous across the optical shadow boundaries. Attention is focussed on some UTD solutions developed recently at the O.S.U. ElectroScience Laboratory; e.g., for the problems of EM diffraction by perfectly-conducting edges, vertices and convex surfaces, and also by a thin dielectric half plane. A few selected applications of these UTD solutions to predict the EM radiation and scattering from complex structures such as aircraft, missile or spacecraft shapes are illustrated.