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The task of modeling and reasoning about real-world problems often involves analyzing overconstrained representations, where not all constraints of a problem can be simultaneously satisfied. The need to analyze over-constrained (or unsatisfiable) problems occurs in many settings, including data and knowledge bases, artificial intelligence, applied formal methods, operations research and description logics. In most cases, the problem to solve is related with some form of minimal unsatisfiability, i.e. an irreducible set of constraints that explains unsatisfiability. This paper provides an overview of some of the computational problems related with minimal unsatisfiability in Boolean logic, including the identification of one minimal unsatisfiable sub-formula and the identification of all minimal unsatisfiable sub-formulas. In addition, the paper briefly overviews practical applications of minimal unsatisfiability. Finally, the paper highlights recent work on minimal unsatisfiability in other domains.