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This work is concerned with efficient techniques for adaptive interference cancellation, and their application to very large arrays. Such arrays provide many degrees of freedom (DOFs) that can be used for filtering. However, the processing hardware required to fully utilize these DOFs (i.e., in a simultaneous, data-adaptive fashion) is prohibitive. To address this problem, we establish a new signal processing framework called "closed-loop multistage adaptive beamforming." A simplistic environmental model is used to demonstrate the capabilities of this new class of signal processing architectures. In typical applications, closed-loop multistage adaptive beamforming is shown to potentially result in one of the following desirable results: (1) orders of magnitude reductions in hardware and processing complexity for a given performance level; and (2) orders of magnitude improvements in interference rejection and target detection for a given number of instantaneously adaptive degrees of freedom.