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It is supposed that there is a multisensor system in which each sensor tests a finite number of hypotheses in a sequential manner. Then the decisions are transmitted to a fusion center, which combines them to improve the performance of the system. First we propose a local multihypothesis sequential test procedure which allows one to fix the probabilities of errors at specified levels and is asymptotically optimal for general statistical models in the sense of minimizing the expected sample size when the probabilities of errors vanish. We then construct two fusion rules-non-sequential and sequential. The first fusion rule waits until all the local decisions in all sensors are made and then fuses them. It is optimal in the sense of minimizing the average probability of error (Bayes criterion) or the maximal probability of error (minimax criterion). In contrast, the sequential fusion rule fuses local decisions one by one in the order they are made, and at each stage decides whether to continue fusion or to stop and make a final decision. It has an advantage over the first rule in that it reduces the total time to make a final decision, for a given average probability of error. An example of fusion of binary local decisions shows that the final decision can be made substantially more reliable even for a small number of sensors (3-5).