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Alpha-stable distributions have been recognized in the signal processing community as simple, yet very accurate, two-parameter statistical models for signals and noise that contain an impulsive component of various degrees of severity. On the basis of this finding, several signal processing problems have been addressed and solved within the framework of alpha-stable distributions and with the use of fractional, lower-order statistics. We attempt to popularize these new signal processing tools within the radar community. In particular, we evaluate the goodness-of-fit of alpha-stable models in the radar environment and test the performance of new signal processing algorithms for signal detection and classification on real radar, sea-clutter data.