The resonance scattering theory (RST) and the singularity expansion method (SEM) are both based on the complex-frequency poles of the scattering amplitude in the scattering of acoustic, elastic, or electromagnetic waves from elastic or impenetrable objects, or from cavities. These poles, situated off the real frequency axis at locations with negative imaginary parts, are found to yield, at the real frequencies of the experiments, prominent resonances for acoustic and elastic-wave scattering from elastic objects as discussed in our earlier review (Überall et al, Appl Mech Rev43 (10), 1990, 235). However, as the authors demonstrated before (Überall et al, J Acoust Soc Am61, 1977, 711), the origin of these resonances lies in the phase matching of circumferential or surface waves generated on the target objects during the scattering; hence a study of the resonances will lead to an understanding of, and information on these surface waves. This has been the topic of a large number of studies in recent years, and the results are summarized in the present review for immersed elastic target objects of plane, spherical, and cylindrical geometry, including both elastic-type and fluid-borne surface waves. For multilayered elastic structures, we also describe possible layer-resonance identifications based on acoustic and elastic-wave scattering experiments.