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Afriat’s Theorem (1967) states that a dataset can be thought of as being generated by a consumer maximizing a continuous and increasing utility function if and only if it is free of revealed preference cycles containing a strict relation. The latter property is often known by its acronym, GARP (for generalized axiom of revealed preference). This paper surveys extensions and applications of Afriat’s seminal result. We focus on those results where the consistency of a dataset with the maximization of a utility function satisfying some property can be characterized by a suitably modified version of GARP.