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Wavelets have been used extensively for several years now in astronomy for many purposes, ranging from data filtering and deconvolution to star and galaxy detection or cosmic-ray removal. More recent sparse representations such as ridgelets or curvelets have also been proposed for the detection of anisotropic features such as cosmic strings in the cosmic microwave background. We review in this paper a range of methods based on sparsity that have been proposed for astronomical data analysis. We also discuss the impact of compressed sensing, the new sampling theory, in astronomy for collecting the data, transferring them to earth or reconstructing an image from incomplete measurements.