  We introduce the notion of generalized bialgebra, which includes the
classical notion of bialgebra (Hopf algebra) and many others. We prove that,
under some mild conditions, a connected generalized bialgebra is completely
determined by its primitive part. This structure theorem extends the classical
Poincar\'e-Birkhoff-Witt theorem and the Cartier-Milnor-Moore theorem, valid
for cocommutative bialgebras, to a large class of generalized bialgebras.
Technically we work in the theory of operads which permits us to give a
conceptual proof of our main theorem. It unifies several results, generalizing
PBW and CMM, scattered in the literature. We treat many explicit examples and
suggest a few conjectures.
