A choice function is defined to exist if there is a "best" (under a binary relation R) element in all non-empty compact subsets of S, the set of all possible alternatives, whereas a demand correspondence exists if there is a "best" element in only the budget sets of S. Some basic restrictions on R are considered. First, if the "at least as good as" sets are closed, then none of the standard restrictions on R are shown to be necessary for the existence of a demand correspondence: the "domination" of finite sets is necessary and sufficient. This is shown to imply that acyclicity of R is necessary and sufficient for the existence of choice functions. Second, if either there is a restriction on convergent P monotone sequences or if R satisfies a regularity condition, then a condition on cyclical sets of alternatives is enough to guarantee the existence of demand correspondences. For the existence of rational choice functions, however, reflexivity, completeness, and transitivity of R, together with the above-mentioned condition on P-monotone sequences, are necessary and sufficient. Finally, if the strictly preferred sets are taken to be convex, then under a restriction weaker than the first, a best element in budget sets exists.