Let G be a finite group generated by a collection S of subsets of G. Define the width of G with respect to S to be the minimal integer n such that G is equal to the union of a product of n subsets in S, together with all subproducts. For example, when S consists of a single subset, the width is just the diameter of the Cayley graph of G with respect to this subset. This article contains a discussion of a variety of problems concerning the width of simple groups, mainly in the following cases: (1) the case where S consists of a single subset; (2) the case where S is closed under conjugation. There are many examples of special interest. Particular emphasis is given to recent results and problems concerning the "word width" of simple groups - namely, the width in the case where S consists of all values in G of a fixed word map. Also discussed are combinatorial interpretations of some width problems, such as the estimation of diameters of orbital graphs.
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