In this case one calls $B G$ a classifying space for $G$-principal bundles.

The universal principal bundle is characterized, up to equivalence, by its total space $E G$ being contractible.

More generally, we can ask for a universal bundle for numerable bundles, that is principal bundles which admit a trivialisation over a numerableopen cover. Such a bundle exists, and classifies numerable bundles over all topological spaces, not just paracompact spaces or CW-complexes.