We show that Friedman's proof of the existence of non-trivial βη-complete models of λ→ can be extended to system F. We isolate a set of conditions that are sufficient to ensure βη-completeness for a model of F (and α-completeness at the level of types), and we discuss which class of models we get. In particular, the model introduced in Barbanera and Berardi (1997), having as polymorphic maps exactly all possible Scott continuous maps, is βη-complete, and is hence the first known complete non-syntactic model of F. In order to have a suitable framework in which to express the conditions and develop the proof, we also introduce the very natural notion of ‘polymax models’ of System F.