Caustics formed in the field diffracted by defects are explored theoretically as a possible approach to the inverse scattering problem for ultrasonic non-destructive evaluation. The case of crack-like defects is considered in detail using the geometrical theory of diffraction. The involute of the far-field caustic reproduces the projection of the crack edge in the incident beam direction, for a plane incident wavefront. This purely geometrical inversion is carried out uniquely for the asteroid and its involute, the elliptical edge. For a general edge shape, the complete inversion requires one further length measurement, which may be carried out in some cases by further experiments with caustics. Useful limitations on the possible shapes of caustics are explained on the basis of catastrophe theory.