We introduce the notion of generalized bialgebra, which includes

the classical notion of bialgebra (Hopf algebra) and

many others, like, for instance, the tensor algebra equipped

with the deconcatenation as coproduct. We prove that, under

some mild conditions, a connected generalized bialgebra

is completely determined by its primitive part. This structure

theorem extends the classical Poincaré-Birkhoff-Witt

theorem and Cartier-Milnor-Moore theorem, valid for co-commutative

bialgebras, to a large class of generalized bialgebras.

Technically we work in the theory of operads which

allows us to state our main results and permits us to give it

a conceptual proof. A generalized bialgebra type is determined

by two operads: one for the coalgebra structure C ,

and one for the algebra structure A. There is also a compatibility

relation relating the two. Under some conditions,

the primitive part of such a generalized bialgebra is an algebra

over some sub-operad of A , denoted P. The structure

theorem gives conditions under which a connected generalized

bialgebra is cofree (as a connected C -coalgebra) and

can be re-constructed out of its primitive part by means of

an enveloping functor from P -algebras to A -algebras. The

classical case is ( C, A, P ) = ( Com, As, Lie ). This structure

theorem unifies several results, generalizing the PBW and

the CMM theorems, scattered in the literature. We treat

many explicit examples and suggest a few conjectures.