This paper considers demand systems for utility-maximizing consumers equipped with additive linearly perturbed utility of the form U(x)+m⋅x and faced with general budget constraints x 2 B. Given compact budget sets, the paper provides necessary as well as sufficient conditions for a demand generating function to be consistent with utility maximization. Within a budget, the convex hull of the demand correspondence is the subdifferential of the demand generating function. The additive random utility discrete choice model (ARUM) is a special case with finite budget sets where utility is considered as random and perturbed by additive location shifters m. Any ARUM can be represented by a choice-probability generating function (CPGF) and every CPGF is consistent with an ARUM. The choice probabilities from the ARUM are the gradient of the CPGF. The paper relates CPGF to multivariate extreme value distributions, and reviews and extends methods for constructing CPGF for applications. The results for ARUM are extended to competing risk survival models.