This article was reprinted in G. Shafer and J. Pearl (eds.), Readings in Uncertain Reasoning, 1990, pp. 575-610, Morgan Kaufmann, San Mateo, CA. Also, a condensed 8-pp version of this paper appeared in the Proceedings of the Fourth Workshop on Uncertainty in Artificial Intelligence in 1988.In this paper, we describe an abstract framework and axioms under which exact local computation of marginals is possible. The primitive objects of the framework are variables and valuations. The primitive operators of the framework are combination and marginalization. These operate on valuations. We state three axioms for these operators and we derive the possibility of local computation from the axioms. Next, we describe a propagation scheme for computing marginals of a valuation when we have a factorization of the valuation on a hypertree. Finally we show how the problem of computing marginals of joint probability distributions and joint belief functions fits the general frameworkResearch for this article has been partially supported by NSF grant IRI-8902444 and a Research Opportunities in Auditing grant 88-146 from the Peat Marwick Foundation's Research Opportunities in Auditing program.