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In many cases, observed brain signals can be assumed as the linear mixtures of unknown brain sources/components. It is the task of blind source separation (BSS) to find the sources. However, the number of brain sources is generally larger than the number of mixtures, which leads to an underdetermined model with infinite solutions. Under the reasonable assumption that brain sources are sparse within a domain, e.g., in the spatial, time, or time-frequency domain, we may obtain the sources through sparse representation. As explained in this article, several other typical problems, e.g., feature selection in brain signal processing, can also be formulated as the underdetermined linear model and solved by sparse representation. This article first reviews the probabilistic results of the equivalence between two important sparse solutions - the 0-norm and 1-norm solutions. In sparse representation-based brain component analysis including blind separation of brain sources and electroencephalogram (EEG) inverse imaging, the equivalence is related to the recoverability of the sources. This article also focuses on the applications of sparse representation in brain signal processing, including components extraction, BSS and EEG inverse imaging, feature selection, and classification. Based on functional magnetic resonance imaging (fMRI) and EEG data, the corresponding methods and experimental results are reviewed.