CLEAN is an algorithm for the suppression of artifacts from nonuniform sampling, originally developed in the field of radioastronomy in the 1960s and 1970s. Recognizing similarities between the problem of NMR NUS and astronomical data collection, several NMR groups have applied versions of CLEAN to NMR data. Here, we recount the historical background of CLEAN in astronomy, explain how the algorithm works as applied to NMR, examine the method's theoretical underpinnings, and present examples of its use with NMR experiments. CLEAN shows a number of similarities to methods being used in the compressed sensing community, and recent theoretical results suggest that CLEAN should succeed under the same or similar conditions to those in which convex l₁-norm minimization succeeds.