We introduce a constructive, incremental learning system for regression problems that models data by means of spatially localized linear models. In contrast to other approaches, the size and shape of the receptive field of each locally linear model, as well as the parameters of the locally linear model itself, are learned independently, that is, without the need for competition or any other kind of communication. Independent learning is accomplished by incrementally minimizing a weighted local cross-validation error. As a result, we obtain a learning system that can allocate resources as needed while dealing with the bias-variance dilemma in a principled way. The spatial localization of the linear models increases robustness toward negative interference. Our learning system can be interpreted as a nonparametric adaptive bandwidth smoother, as a mixture of experts where the experts are trained in isolation, and as a learning system that profits from combining independent expert knowledge on the same problem. This article illustrates the potential learning capabilities of purely local learning and offers an interesting and powerful approach to learning with receptive fields.