The singularity expansion method (SEM) for quantifying the transient electromagnetic (EM) scattering from targets illuminated by pulsed EM radiation is reviewed. SEM representations for both induced currents and scattered fields are presented. Natural-resonance-based target identification schemes, based upon the SEM, are described. Various techniques for the extraction of natural-resonance modes from measured transient response waveforms are reviewed. Particular attention is given to the aspect-independent (extinction) E-pulse and (single-mode) S-pulse discriminant waveforms which, when convolved with the late-time pulse response of a matched target, produce null or mono-mode responses, respectively, through natural-mode annihilation. Extensive experiment results for practical target models are included to validate the E-pulse target discrimination technique. Finally, anticipated future extensions and areas requiring additional research are identified.>