We consider a broad class of situations where a society must choose from a finite set of alternatives. This class includes, as polar cases, those where the preferences of agents are completely unrestricted and those where their preferences are single-peaked. We prove that strategy-proof mechanisms in all these domains must be based on a generalization of the median voter principle. Moreover, they must satisfy a property, to be called the "intersection property," which becomes increasingly stringent as the preference domain is enlarged. In most applications, our results precipitate impossibility theorems. In particular, they imply the Gibbard-Satterthwaite theorem as a corollary.Journal of Economic LiteratureClassification Number: D71. © 1997 Academic Press.