Recent years have demonstrated that ultrasonic waves in solids permit investigations of wave chaos, reverberation and multiple scattering that complement those of microwaves, optics and electronics. The large quality factors that can be achieved with elastic waves, their relatively slow wave speed, and their good impedance mismatch with the laboratory, lead to certain advantages for the experimentalist and new perspectives for the theorist. In particular these features allow transients to be observed, allow good isolation from leads and antennas, and convenient time and length scales in the experiment. A short course on continuum elastodynamics in solids is followed by a review of the laboratory work on spectral and transport properties of diffuse ultrasonic waves in solids. These include demonstrations of energy equipartition, of eigenvalue correlations and spectral rigidity, enhanced backscatter in enclosures, wave diffusion, and Anderson localization.