A neural network approach has been developed to determine the depth of a surface breaking crack in a steel plate from ultrasonic backscattering data. The network is trained by the use of feedforward three-layered network together with a back-scattering algorithm for error correction. The signal used for crack insonification is a mode converted 70$^{circ}$transverse wave. A numerical analysis of back scattered field is carried out based on elastic wave theory, by the use of the boundary element method. The numerical data are calibrated by comparison with experimental data. The numerical analysis provides synthetic data for the training of the network. The training data have been calculated for cracks with specified increments of the crack depth. The performance of the network has been tested on other synthetic data and experimental data which are different from the training data.