By unification, we understand a family of algorithms employed by compu tational versions of certain grammar formalisms to combine information in feature structures. The use of these formalisms has become widespread, and several extensions to the basic notion of feature structure have been proposed. Although algorithms for unification of these extended feature structures have been written, they are complicated, and a precise model of feature structures is desirable to give an adequate specification of what the algorithms do. We have developed a model in which descriptions of feature structures can be regarded as logical formulas, and interpreted by sets of directed graphs which satisfy them. We identify a feature structure with such a directed graph, but our mathematical work is facilitated by considering the graphs to be transition graphs for a special type of deter ministic finite automaton. This semantics for feature structures extends the ideas of Pereira and Shieber [11], by providing a way to model feature values which are speci fied by disjunctions and non-local path values embedded within disjunc tions. Our interpretation differs from that of Pereira and Shieber by using a logical model in place of a denotational semantics. The model yields a calculus of equivalences between formulas, which can be used to simplify them. A similar use of logic to describe feature structures was first pro posed in Generalized Phrase Structure Grammar (GPSG)[3], in order to describe feature co-occurrence restrictions. Our formulation, which was developed independently, is for the purpose of understanding the proper ties of unification. Recently Gazdar et al. [2] have extended their logic in order to give uniform descriptions of linguistic categories. The two logics have much in common when presented formally, and it should be possible to combine them in a uniform way. This, however, will be left for future work.