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A basic phenomenon in random structures such as random graphs is the threshold phenomenon, where a system undergoes a swift qualitative change as result of a small change in a parameter guiding its probabilistic structure. In an earlier paper [J Amer Math Soc 12 (1999), 1017–1054] a general criterion was presented for structures to undergo such a phase transition. In this paper we give a survey of the state of the art in applying the aforementioned criterion, exemplify the techniques by proving the existence of a sharp threshold for hypergraph colorability, and present some related open problems. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 26, 2005Research supported in part by the Israel Science Foundation, Grant No. 0329745.