In this chapter, I explore the property law treatment of cryptoassets—and, presumably, other digital artefacts that are not clearly rights to anything or against anyone. I argue that such artefacts are well described as “ideational objects” and draw together insights from private law theory and social ontology to explore how we should think about complex objects with a social and a technical dimension. I then examine how the law of property can accommodate ideational objects as objects of property rights (including the right of ownership) and dealing such as transfer, and what challenges and implications this poses for the traditional doctrines and concepts of property law. The chapter aims to contribute to the emerging literature on cryptoassets (and digital assets more broadly), as well as to the broader debate on the nature and boundaries of property in the digital age. In particular, it defends the Law Commission’s suggested approach of recognizing a “third category” of personal propertyPersonal property from criticisms made by Professor Kelvin Low, and introduces a new conceptual framework from Professor Christian von Bar’s seminal treatise on “things as objects of property rights”.