A set theory with the axioms of Zermelo set theory (Extensionality, Union, Pair-set, Foundation, Restriction, Infinity, Power-set) plus the Replacement axiom schema:
If F(x,y) is a formula such that for any x, there is a unique y making F true, and X is a set, then
F x : x in X
is a set. In other words, if you do something to each element of a set, the result is a set.
An important but controversial axiom which is NOT part of ZF theory is the Axiom of Choice.