Applications
Aczel's hypersets were extensively used by Jon Barwise and John Etchemendy in their 1987 book The Liar, on the liar's paradox; The book is also good introduction to the topic of non-well-founded sets.
Boffa's superuniversality axiom has found application as a basis for axiomatic nonstandard analysis.
Read more about this topic: Non-well-founded Set Theory