We here, give a brief survey of our new many-body theory for composite excitons, as well as some of the results we have already obtained using it. In view of them, we conclude that, in order to fully trust the results one finds, interacting excitons should not be bosonized: indeed, all effective bosonic Hamiltonians (even the hermitian ones!) can miss terms as large as the ones they generate; they can even miss the dominant term, as in problems dealing with optical nonlinearities. (c) 2004 Elsevier Ltd. All rights reserved.