Work in both animals and humans has emphasised that the brain specifically tracks the space near the body---the so-called peripersonal space (PPS). These representations appear to be multimodal and expressed in body-centred coordinates. They also play an important role in defence of the body from threat, manual action within PPS, and the use of tools---the latter, notably, 'extending' PPS to encompass the tool itself. Yet different authors disagree about important aspects of these representations. As de Vignemont and Iannetti noted, it is unclear whether there is one representation or multiple functionally distinct representations. It is also unclear whether what is represented is the body, objects, relationships between the two, or some more abstract set of properties---and introspection is curiously silent on the matter. I suggest  a mildly deflationary position: areas which subserve PPS are primarily devoted to complex, context-sensitive coordinate transformations. This is still (I argue) a representational relationship, and preserves other important representational relationships, but is distinct from the sorts of representations that generate phenomenal awareness. The deflationary position also explains some curious linkages and dissociations between the perception of pain, threat, and environmental affordances.