The article presents an overview of transient resonance scattering, emphasizing one of its most important applications—the active classification of sonar and radar targets. It discusses traditional, classical techniques such as the Watson-Sommerfeld method (WSM) to transform classical, and slowly convergent normal-mode series in the frequency domain, to rapidly convergent series in the domain of the complex generalization, λ, of the mode-order, n. In view of its analytical complexity and the advent of computers that can overcome slow convergence difficulties, the WSM is not as popular today as it once was. Its main advantage remains its ability to extract physical interpretations from the mathematical results. Resonance scattering focuses on the resonance spectral region of targets. Of these, the penetrable (ie, elastic or dielectric) ones are the subjects of main interest here, particularly those insonified/illuminated by (finite) pulses of various types. The authors describe the exact isolation and extraction of the resonances contained within the scattering cross-section of a penetrable target by subtraction of suitable, background, geometrical contributions. These backgrounds are often given by the solution for an identical, but impenetrable target. This seems to be the main usefulness of impenetrable target solutions in underwater acoustics, which, generally, are physically unrealistic idealizations. The resonances identify the target as its fingerprint. Examples are shown to illustrate various transient scattering phenomena in acoustics and electromagnetism. The article shows exactly how the broadband pulses emitted by an impulse sonar (or radar) extract a substantial number of resonances from the echoes of penetrable targets. Further, it is shown how these are actually used to identify all physical characteristics of various analyzed targets, thus, indeed identifying them. The application of a novel signal processing technique that analyzes the echoes in the joint time-frequency domain is examined. This shows much promise for target identification purposes. Many distributions of the Wigner-type were used by us to generate simulated and experimental echo-displays in time-frequency that show the advantages of the process. The present overview supplements two earlier ones [23, 48] on closely related subjects. The article includes 101 references.