We characterize strategy-proof social choice functions when individuals have strictly quasi-concave, continuous and satiated utility functions on convex subsets of IR m, representing preferences for the provision of m pure public goods. When specialized to the case m = 1, these assumptions amount to requiring that preferences are single peaked, and for such a domain there exists a wide class of strategy-proof social choice functions. These were studied by Moulin (1980) under strong additional assumptions. Our first results characterize the complete class, after an appropriate extension of the single-peakedness condition. The new characterization retains the flavour of Moulin's elegant representation theorem. For the general m-dimensional case, previous results have shown that there is no efficient, strategy-proof, nondictatorial social choice function, even within the domain restrictions under consideration (Border and Jordan 1983;Zhou 1991). In fact, Zhou's powerful result indicates that nondictatorial strategy-proof s.c.f.'s will have a range of dimension one. This allows us to conclude with a complete characterization of all strategy-proof s.c.f.'s on IR m, because restrictions of preferences from our admissible class to one dimensional subsets satisfy the slightly generalized notion of single-peakedness that is used in our characterization for the case m = 1. We feel that a complete knowledge of the class of strategy-proof mechanisms, in this as well as in other contexts, is an important step in the analysis of the trade-offs between strategy-proofness and other performance criteria, like efficiency. This paper was written while both authors were visiting GREMAQ, Universitd des Sciences Sociales de Toulouse. We are thankful for its hospitality and good research atmosphere. Bar-berA's work is supported by the Instituto de Estudios Fiscales and by research grant PB89-0294 from the Secretaria de Estado de Universidades e Investigacidn, Spain. Jackson acknowledges the support of NSF grant SES8921409. We thank Jacques Cr6mer, Beth Allen, John Weymark and two anonymous referees for helpful comments on earlier drafts.