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A tutorial-style framework is presented for understanding the current status of adaptive infinite-impulse-response (IIR) filters. The paper begins with a detailed discussion of the difference equation models that are useful as adaptive IIR filters. The particular form of the resulting prediction error generic to adaptive IIR filters is highlighted and the structures of provable convergent adaptive algorithms are derived. A brief summary of particular, currently known performance properties, drawn principally from the system identification literature, is followed by the formulation of three illustrative adaptive signal processing problems, to which these adaptive IIR filters are applicable. The concluding section discusses various open issues raised by the formulation of this framework.