It is well known that in the scattering of elastic waves from localized inhomogeneities the scattering amplitude2A is proportional to the square of the frequency win the Rayleigh {long wavelength) regime, i.e., A= A2w + ... This talk deals with the problem of (1) extracting A2 from experimental scattering data, (2) calculating A2 for an assumed scatterer and (3) deducing the properties of the scatterer from a set of values of A2 measured for various transducer configurations. A review of experimental and theoretical results for A2 will be presented for the case.of spheroidal voids and the remaining discrepancies between the two kinds of results will be discussed. The inverse problem (i.e., deducing the scatterer properties from the scattering measurements) will be discussed in detail. The probabilistic inverse problem, which provides the appropriate framework for the interpretation of real data, will be covered at greater length. In the case in which it is assumed that the scatterer is an ellipsoid void, \those size, shape and orientation are unknown a priori, a number of computational results involving best estimates and associated measures of significance will be given. Analogous results will be derived for parameters related to fracture mechanics.