Statistical models defined by imposing restrictions on marginal distributions of contingency tables have received considerable attention recently. While these models are flexible and useful, certain theoretical questions have remained open in the literature. These include, firstly, the existence of a joint distribution with certain restrictions on some of its marginals, or general conditions under which the existence of such distributions is guaranteed; secondly, the determination of the dimension of a model; thirdly, a formal argument for the applicability of large sample results for maximum likelihood estimates. A general theory for answering these questions is discussed in this article.