The chapter describes an important class of the nonlinear adaptive system commonly known as artificial neural networks or just simply neural networks. A neural network is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use. The chapter presents the use a filter to perform three basic information-processing tasks: filtering, smoothing, and prediction. A filter is said to be linear if the filtered, smoothed, or predicted quantity at the output of the device is a linear function of the observations applied to the filter input. Otherwise, the filter is nonlinear. A wide variety of recursive algorithms have been developed in the literature of the operation of linear adaptive filters. The chapter describes a spatial form of adaptive signal processing that finds practical use in radar, sonar, communications, geophysical exploration, astrophysical exploration, and biomedical signal processing.