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This paper considers series-cascade nonlinear adaptive filter architectures consisting of a linear input filter, a memoryless polynomial nonlinearity, and a linear output filter (LNL). The learning characteristics of the LNL structure are studied in terms of performance and complexity. Replacing the linear input stage and the memoryless nonlinear stage of the LNL model with a Volterra module is then considered. Adaptive algorithms are summarized for these structures and experimental examples are used to illustrate performance for the identification of an acoustic echo channel.