This paper describes and discusses the history and state of the art of continuous backbone robot manipulators. Also known as continuum manipulators, these robots, which resemble biological trunks and tentacles, offer capabilities beyond the scope of traditional rigid-link manipulators. They are able to adapt their shape to navigate through complex environments and grasp a wide variety of payloads using their compliant backbones. In this paper, we review the current state of knowledge in the field, focusing particularly on kinematic and dynamic models for continuum robots. We discuss the relationships of these robots and their models to their counterparts in conventional rigid-link robots. Ongoing research and future developments in the field are discussed.