IN THIS PAPER we consider procedures for going from several individual preferences among several alternatives, called candidates, to something which may be called a collective preference. The individual preferences take the form of (total) orderings of the alternatives, and the collective preference is to take the form of a (total) weak ordering (i.e., ties allowed). We consider certain properties which seem desirable in such systems and investigate which systems have these properties. The point of view taken here differs from that of other work in this area (e.g., [1, 2, 3, 4]) chiefly in asking that the procedure work for all possible sizes of the voting population, rather than for a fixed population, given in advance. This permits us to require, for example, that if each of two bodies of voters prefers candidate A to candidate B under a given procedure, then the combination of these bodies should prefer A to B under the same procedure. In Section 1 we give the formal definitions of an aggregation procedure and discuss certain desirable features, namely "neutrality" (treats candidates symmetrically), "separability" (the condition mentioned above), "monotonicity," and an "Archimedean" property which says, roughly, that a sufficiently large body with a given distribution of preferences can impose its will on any body of fixed size. In Section 2 we introduce certain procedures: "point systems" and "generalized point systems" (roughly, point systems allowing "infinitesimal points"), which are neutral and separable. They are monotonic if and only if the points are arranged in the "natural" order, and the point systems are, in addition, Archimedean. In Section 3 we prove a converse, namely that any neutral and separable procedure can be realized by a generalized point system and, if it is Archimedean, by a point system. This part requires some familiarity with the notions of least upper bound of a set of real numbers and bases of vector spaces. In Section 4 (which is largely independent of Section 3), we consider "point runoff" systems which use point systems in a succession of eliminations. Such systems are neither separable nor monotonic but do satisfy some very weak separability and monotonicity conditions. While these probably do not characterize point runoff systems, we know of no other systems satisfying them.