TITLE: Alternative set theories
QUESTION [26 upvotes]: This is a (soft!) question for students of set theory and their teachers. 
OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student encounter (at least to the extent of knowing that the theory exists, and about why it is thought interesting)?
Suggestions might be SP (Scott-Potter), NBG, NF, ZFA, IST, ETCS (for a few sentences of explanation, see http://www.logicmatters.net/2013/03/tyl-14-alternative-set-theories/
But what would be on your list of alternative set theories would be it good for advanced students to hear about, if only briefly?

REPLY [2 votes]: I presented in my web site an expression of set theory that seems new. My purpose in designing it, was to give a clean and efficient way to build mathematics from scratch for the interest of clever undergraduate students, providing a direct path to higher mathematical tools and meta-mathematical considerations.
This is not only a different formalization of set theory but a way to introduce mathematics combining this set theory with model theory.
To sum up the main differences of this set theory from ZF:
The basic kinds of objects are elements, sets and functions. Each element is either a set or a function or none (pure). More kinds of objects can either be seen as additional kinds or constructed from the previous ones : operations, relations, tuples.
Many symbols are admitted beyond the mere ∈ of ZF, most of which are not new but an official recognition of widely used mathematical notations, here taken as (more or less) primary. Some are in neither of both standard kinds of structure symbols of first-order logic (operators and predicates) but are a different kind of symbols: the binders, that bind a variable on a set. Apart from including bounded quantifiers as particular cases, examples include the function definer (E∋x↦…); the set-builders {x∈E|…} and {f(x)|x∈E}; symbols of union and product of a family of sets defined by a term. Also I express and justify a principle giving several axioms of set existence as particular cases.
Classes are defined not as objects of this set theory but as meta-objects.