This review deals with the utility, scope, performance, and range of validity of the discretized Gabor-based, quasi-ray, narrow-waisted (NW) Gaussian beam (GB) algorithm for the analysis and synthesis of high frequency time-harmonic as well as short-pulse transient electromagnetic wavefields in the presence of complex propagation and scattering environments. Restricting attention here primarily to two-dimensional (2-D) fields and physical configurations, applications include phased and focused truncated plane-aperture-generated illumination of layered dielectrics, moderately rough air-soil interfaces, and buried objects in rough-surface-bounded halfspaces in forward scattering scenarios, as well as rough interface profile reconstruction and buried-target imaging from sparse data in inverse scattering scenarios. The role of the Gabor-based NW-GB algorithm as a computationally efficient physically incisive analytic forward solver in these applications is emphasized. Current status is reviewed and assessed in detail, with brief discussion of plans for future extensions, and of recently developed alternative methodologies.