Artificial neural network applications are usually tuned up by experimentation. Several combinations of network parameters and size of the training set are evaluated until a reasonable good performance of the network is achieved. Our prediction of the behaviour of a neural network is so weak, that frequently there is no other choice than to adjust its parameters by trial and error. That is an approach we wished to be part of the past. In this dissertation we should characterize the learning process of a neural network as the probability of generation of the correct (or near correct) output, for samples never seen during the training phase. Statistical Learning Theory and Probably Approximately Correct learning provide the basic model for reasoning about measurable properties of a learning machine. Sample complexity, that is, training set size, machine size, training error, generalization error, and confidence, are woven in a single equation to describe the learning properties of a learning machine. We derive theoretical bounds to the sample complexity and generalization of linear basis neural networks, and radial basis neural networks