While most of the axioms of ZFC assert the existence of certain sets, the axiom of foundation is one of the few axioms which limit the type of sets which can exist. The axioms says (For all A not the emptyset) (there exists B in A) such that (For all C in A) C is not in B In other words this axiom guarantees that all sets are "wellfounded". While this axiom is incredibly useful for studying the properties of a universe of sets, from a naïve set theoretic point of view it is not clear why ill-founded sets shouldn't be able exist. As such, one might hope that by allowing ill-founded sets we could better express the naïve set theoretic idea that "all sets that can exist do". However, when we do this we find that we very quickly run into a problem. When we remove the axiom of foundation, the axiom of extensionality looses a lot of its power. The axiom of extensionality says that two sets are equal if and only if they contain the same elements. One of the most useful consequences of foundation and extensionality is that every set is uniquely determined by its structure. However, once we remove regularity this no longer holds. (To see this consider two sets x = { x } and y = { y }. These sets obviously have the same structure but are not necessarily the same) The Aczel's anti-foundation axiom is an attempt to handle ill-founded sets while still keeping the important property that a set is defined by its structure. In this talk we will introduce Aczel axiom and discuss how it ensures that a set is determined by it's structure as well as how it relates to the axiom of extensionality.