Recent and remarkable advances in the experimental study of acoustic scattering from targets immersed in water are leading to a new spectroscopy: resonance acoustic spectroscopy. The discovery and improvement of an intriguing method, the Method of Isolation and Identification of Resonances (MIIR), has made possible experimental determination of the eigenfrequency spectra of aluminum-elastic cylinders and cylindrical shells. This method gives a quasilinear “resonance spectra.” In addition, it shows the importance of circumferential waves which generate standing waves. They allow us to explain the “reradiation” of targets after the end of insonification. The MIIR has numerous applications, especially in “underwater acoustics” and “nondestructive testing.”