A Wave-Net is an artificial neural network with one hidden layer of nodes, whose basis functions are drawn from a family of orthonormal wavelets. The good localization characteristics of the basis functions, both in the input and frequency domains, allow hierarchical, multiresolution learning of input-output maps from experimental data. Furthermore, Wave-Nets allow explicit estimation for global and local prediction error-bounds, and thus lend themselves to a rigorous and explicit design of the network. This article presents the mathematical framework for the development of Wave-Nets and discusses the various aspects of their practical implementation. Computational complexity arguments prove that the training and adaptation efficiency of Wave-Nets is at least an order of magnitude better than other networks. In addition, it presents two examples on the application of Wave-Nets; (a) the prediction of a chaotic time-series, representing population dynamics, and (b) the classification of experimental data for process fault diagnosis.