A universal -principal bundle is a -principal bundle, which is usually written , such that for every CW-complex the map
[X, B G] \to G Bund(X)/_\sim
In this case one calls a classifying space for -principal bundles.
The universal principal bundle is characterized, up to equivalence, by its total space being contractible.
More generally, we can ask for a universal bundle for numerable bundles, that is principal bundles which admit a trivialisation over a numerable open cover. Such a bundle exists, and classifies numerable bundles over all topological spaces, not just paracompact spaces or CW-complexes.
Among the earliest references that consider the notion of universal bundles is
A review is for instance in