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An exhaustive review of adaptive signal processing schemes which are robust, sparsity-aware and robust as well as sparsity-aware has been carried out in this paper. Conventional robust learning approaches as well as the ones based on information theoretic methods have been included in the review. Further, adaptive filtering schemes which take advantage of the sparse nature of the system impulse responses have been reviewed, including the ones which are also robust. The cost functions used in these algorithms have been summarized and a timeline of algorithm development in this area has been added to provide an excellent overview on the topic. [ABSTRACT FROM AUTHOR]