Mathematically adequate and philosophically principled accounts of proper classes appear to be essential for investigations on the foundations of mathematics. Indeed, some key set-theoretical results, such as Kunen's elementary embedding theorem, crucially require proper classes; also, principled accounts of proper classes can shed light on questions concerning the resolution of paradoxes of self-reference and the connection of class membership with related notions, such as property instantiation. Furthermore, the prospects for a class-theoretic foundations of category theory, as well as the possibility to formulate class-theoretic semantics, and to handle unrestricted quantification, provide us with reasons to believe that a further desideratum for a theory of proper classes should be some form of self-applicability. 
A possible way to obtain a mathematically adequate, philosophically principled theory of type-free classes is to extend theories of non-well-founded sets such as Aczel's (1988): this is, for instance, what Barwise and Moss set out to do, in their book "Vicious Circles" (1996). However, Barwise and Moss's account presents some mathematical and methodological shortcomings. In this talk. I will sketch a proposal for a new conception of type-free classes, which I will call "the closed conception of class", and which extends the conception underlying non-well-founded sets to the class-theoretic realm. The conception sidesteps some issues in Barwise and Moss's proposal, and provides a principled solution to the self-referential paradoxes. Furthermore, the conception is licensed by a family of model constructions called hyperuniverses, which have been studied at length by Forti and Honsell (1989). After developing the conception, I will explore some ideas and prospects towards a principled axiomatisation of closed classes, focusing on obtaining a theory satisfying the desideratum of mathematical adequacy.

- Peter Aczel. Non-Well-Founded Sets. Palo Alto, CA, USA: CSLI Lecture Notes, 1988.
- Jon Barwise and Lawrence Moss. Vicious circles: on the mathematics of non-wellfounded phenomena. Center for the Study of Language and Information, 1996.
- Marco Forti and Furio Honsell. Models of self-descriptive set theories. Springer, 1989.