An outcome is ambiguous if it is an incomplete description of the probability distribution over consequences. An `incomplete description' is identified with the set of probabilities that satisfy the incomplete description. A choice problem is uncertain if the decision maker is choosing between distributions, and is ambiguous if the decision maker is choosing between sets of probabilities. The von Neumann/Morgenstern approach to uncertain choice problems uses a continuous linear function on probabilities. This paper develops the theory of ambiguous choice problems as a continuous, linear functions on closed convex sets of probabilities. This delivers: a framework encompassing most of the extant ambiguity averse preferences; a complete separation of attitudes towards risk and attitudes toward ambiguity; and generalizations of rst and second order stochastic dominance rankings to ambiguous decision problem. Quasi-concave preferences on sets that satisfy a restricted betweenness property capture variational preferences. / text