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Much attention has been given in recent years to the problem of finding Minimally Unsatisfiable Subformulas (MUSes) of Boolean formulas. In this paper, we present a new view of the problem, strongly linking it to the maximal satisfiability problem. From this relationship, we have developed a novel technique for extracting all MUSes of a CNF formula, tightly integrat ing our implementation with a modern SAT solver. We also present another algorithm for finding all MUSes, developed independently but based on the same relationship. Experimental comparisons show that our approach is con sistently faster than the other, and we discuss ways in which ideas from both could be combined to improve further.