Beyond Rigidity is divided into two parts. Part 1 is devoted to the semantics and pragmatics of names, and the sentences, including attitude ascriptions, that contain them. In part 2, the model developed in part 1 is extended to natural kind terms, and simple predicates in which they occur. The model is then used to explain the necessity of certain aposteriori statements containing such predicates. Part 1 begins with a review of Kripke’s arguments against traditional descriptive semantic analyses of proper names. Special attention is paid to the modal argument, which is strengthened to rule out analyses that treat names as synonymous with rigidified descriptions, or descriptions that are required to take wide scope when embedded under modal operators or predicates. I then try to fill an important gap in Kripke’s original story by providing a positive account of the meaning and semantic content of names. The central question is What do proper names contribute to the semantic contents of ( i.e. the propositions semantically expressed by) sentences that contain them?