We initiate the study of perfectly secure multiparty computation over asynchronous networks tolerating generalized adversaries. The classical results in information-theoretically secure asynchronous multiparty computation among n players state that less than (formula presented) active adversaries can be tolerated in the perfect setting [4]. Strictly generalizing these results to the non-threshold setting, we show that perfectly secure asynchronous multiparty computation among n players tolerating the adversary structure A is possible if and only if the union of no four sets in the adversary structure cover the full set of players. The computation and communication complexities of the presented protocols are polynomial in the size of the maximal basis of the adversary structure. Our results generalize the results of [16,10] to the asynchronous setting. Furthermore, when restricted to the threshold setting, the protocols of this paper result in solutions as good as the best known asynchronous threshold protocols for the perfect setting. Incidentally, the problems of designing efficient asynchronous secure protocols and adapting the efficiency improvement techniques of the threshold setting to the nonthreshold setting were mentioned as open in [18,17]. © Springer-Verlag Berlin Heidelberg 2002.