Here we offer a thorough review of the empirical literature on the conditions under which an object, such as a tool or a prosthetic (whether real or virtual), can be experienced as being in some sense a part or extension of one's body. We discuss this literature both from the standpoint of the apparent malleability of our body representations, and also from within the framework of radical embodied cognition, which understands the phenomenon to result not from an alteration to a representation, but rather from the achievement of a certain kind of sensory/motor coupling. We highlight both the tensions between these frameworks, and also areas where they can productively complement one another for future research.