On the basis of work in [8] for nondeterministic left-to-right parsing, we trace here a theory of head-driven parsing going from crude top-down and head-corner to more sophisticated solutions, in the attempt to successively make more deterministic the behaviour of head-driven methods. Finally, we propose an original generalization of headdriven parsing, allowing a more detailed specification of the order in which elements of a right-hand side are to be processed.