New Foundations, the set theory invented by Quine in 1937, has a notoriously strange universe of sets governed by a stratification condition on formulas. While Randall Holmes recently announced a proof of the consistency of NF relative to standard set theory ZFC, the internal structure of NF models remains poorly understood. In particular, it is unknown whether NF can prove or is consistent with the existence of a definable well-ordering of its entire universe, a property that would dramatically clarify the ordinal and combinatorial structure of NF models. The recent direct proof of the Axiom of Infinity in the weakened system NF-0 suggests that elementary combinatorial facts can be established more directly in NF-style theories than previously thought, but the question of global choice and well-ordering in full NF remains unresolved.