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The problem of designing networks that can survive an enemy attack or natural disaster has received considerable attention in recent years. Work in this area has focused on the formulation of survivability criteria, the development of analysis methods to rank networks with respect to these criteria, and the generation of networks which are optimal with respect to these criteria. Many partial results for a variety of network models are available. The purpose of this paper is to summarize the most significant of these results, to link various models and approaches, and to indicate areas of study in which additional research seems both desirable and feasible.