Full Idea
The theory of types is a thing of the past. There is now nothing to choose between ZFC and NBG (Neumann-Bernays-Gödel). NF (Quine's) is a more specialized taste, but is a place to look if you want the universe.
Gist of Idea
Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe
Source
William D. Hart (The Evolution of Logic [2010], 3)
Book Reference
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.88