Syllogisms are arguments about the properties of entities. They consist of 2 premises and a conclusion, which can each be in 1 of 4 "moods": "All A are B," "Some A are B," "No A are B," and "Some A are not B." Their logical analysis began with Aristotle, and their psychological investigation began over 100 years ago. This article outlines the logic of inferences about syllogisms, which includes the evaluation of the consistency of sets of assertions. It also describes the main phenomena of reasoning about properties. There are 12 extant theories of such inferences, and the article outlines each of them and describes their strengths and weaknesses. The theories are of 3 main sorts: heuristic theories that capture principles that could underlie intuitive responses, theories of deliberative reasoning based on formal rules of inference akin to those of logic, and theories of deliberative reasoning based on set-theoretic diagrams or models. The article presents a meta-analysis of these extant theories of syllogisms using data from 6 studies. None of the 12 theories provides an adequate account, and so the article concludes with a guide--based on its qualitative and quantitative analyses--of how best to make progress toward a satisfactory theory. (Contains 8 tables.)