Further results are presented for the direct problem of scattering of high-frequency waves by cracks in elastic solids. Results for a penny-shaped crack, obtained on the basis of geometrical diffraction theory, are compared with experimental data. For simple crack geometries a hybrid method, whereby the crack-opening displacement is computed by ray theory, and the scattered field is subsequently obtained by the use of a representation theorem, is tested by comparison with exact results. The simple form of the far-field high-frequency solutions to the direct scattering problem suggests the application of Fourier-type integrals to solve the inverse problem. Two different inversion integrals are discussed. The inversion method is checked by applying it to the scattered field of a flat elliptical crack, for which an analytical expression is derived. Some computational technicalities are discussed, and numerical results are presented.