This paper describes efficient protocols for multi-party computations that are information-theoretically secure against passive attacks. The results presented here apply to access structures based on quorum systems, which are collections of sets enjoying a naturally-motivated self-intersection property. Quorum-based access structures include threshold systems but are far richer and more general, and they have specific applicability to several problems in distributed control and management. The achievable limits of security in quorum-based multi-party computation are shown to be equivalent to those determined by Hirt and Maurer in [HM97], drawing a natural but non-obvious connection between quorum systems and the extremes of secure multi-party computation. Moreover, for both the general case and for specific applications, the protocols presented here are simpler and more efficient.