Elastic-wave scattering from various types of cavities and inclusions has been studied theoretically with special emphasis on surface wave effects that appear during the scattering process. Resonances in the scattering amplitudes are caused by the phase matching of circumnavigating surface waves, and manifest themselves as poles in the complex frequency plane that correspond to the (complex) eigenfrequencies of the cavity of inclusion. These results are most easily obtained for scatterers of separable geometry, such as spheres, where theoretical amplitudes are well-known. Here, the formalism for a complete treatment of elastic-wave scattering from infinite cylindrical cavities and solid inclusions has been worked out for general oblique incidence. Poles of scattering amplitudes have been found for evacuated and for fluid-filled cylinders, and have been physically interpreted in terms of helical surface waves propagating both interior and exterior to the cylinder. Dispersion, attenuation, and refraction of these surface waves have been obtained. Progressing to more generally-shaped obstacles, we have studied surface waves and complex-frequency poles for finite- length cylindrical cavities with flat ends. In this fashion, the resonance features(particularly the cavity eigenfrequencies) that appear prominently in the scattering amplitude can be understood as to their physical origin and their dependence on the type of cavity, and may be exploited for purposes of classification and identification of flaws by their ultrasonic resonances (ultrasonic “resonance spectroscopy”).