We outline the area of “nonclassical” nonlinear acoustics that deals with media exhibiting anomalously strong elastic nonlinearity, including hysteresis, conditioning, and slow dynamics. The entire range of behaviors we term nonlinear, nonlinear, nonequilibrium dynamics (NND). The link between the diverse materials that exhibit NND appears to be the presence of “damage” at many scales, ranging from order 10−8 m to 10−1 m at least. The “damage” may be distributed as in granular materials or isolated, as in a sample with a single crack. As to the precise physical origin of the behavior, it is relatively clear in granular media for instance (Hertz‐Mindlin interaction physics); however, in other materials the origin is less clear. We believe it to be due to shear sliding, related to crack and possibly dislocation dynamics as well as other, as yet unidentified physical processes. Because the origin of the behavior is related to damage, damage diagnostics in solids, nonlinear nondestructive evaluation, follows naturally. New areas of research and application have appeared recently. For instance, NND plays a significant role in earthquake strong ground motion and potentially to earthquake source dynamics. Medical applications are in development as well as nonlinear imaging methods as well as applications to locating landmines.