The axiomatic method familiar to mathematicians has inspired axiomatic approaches to other domains, including economic and political domains. I will discuss the axiomatic approach to the theory of voting, a topic that is both practically relevant for the design of elections and of interest to theorists in several fields. Although there are differences in how “axioms” are viewed across domains, there are also common types of questions. Given a set of axioms for geometry, set theory, etc., we may ask: does there exist a model that satisfies all of the axioms? Which axioms are independent of others? Can we find axioms that uniquely characterize a given model? Parallel questions arise in the theory of voting. After reviewing the classic axiomatic impossibility theorems of Arrow and Gibbard-Satterthwaite, I will discuss a new impossibility theorem for preferential voting, wherein voters submit rankings of the candidates. Roughly speaking, no voting rule satisfies the following four principles: a candidate who beats all other candidates in head-to-head majority comparisons must win the election; a candidate who loses to all others in head-to-head majority comparisons must lose; adding voters who rank a candidate first should not turn that candidate from a winner into a loser; and ties for winning can be broken by adding a bounded number of voters. Any three of these axioms is satisfied by some voting rule actually in use, whereas the classic impossibilities include axioms violated by almost all voting rules in use. Thus, the new theorem presents a genuine four-way fork in the road for choosing a voting rule. It exemplifies how axiomatic results can be useful in designing a voting rule for a particular application, by making salient the tradeoffs one must make.