Depending on the angle of illumination, electrically large scatterers can support a variety of electromagnetic (EM) phenomena, such as traveling waves, creeping waves, and edge/surface diffraction effects. The electrical size of a body limits the tractability of numerical methods such as the method of moments (MM), and the geometric complexity of an object circumscribes the applicability of optics-derived methods. Hybrid methods incorporating both numerical and high-frequency asymptotic techniques have the potential to substantially enlarge the class of EM scattering problems that can be treated. In this discussion, the current-based hybrid formulation is summarized for classes of two- and three-dimensional scatterers. The use of Ansatz solutions derived from physical optics, the physical theory of diffraction, and the Fock theory is illustrated for perfectly conducting, partially penetrable, and totally coated bodies. For the latter, a generalization rooted in the impedance boundary (Leontovich) condition is used. Complementing these Ansatz solutions, the Galerkin representation is used for regions where the foregoing are computationally or physically intractable. The above cases are illustrated by representative solutions explicating the approach.>