  We introduce Galois Theory for Hopf-Galois Extensions proving existence of a
Galois connection between subalgebras of an H-comodule algebra and generalised
quotients of the Hopf algebra H. Moreover, we show that these quotients Q which
define Q-Galois extension are the closed elements of our Galois connection. We
generalise important results of Hopf--Galois Theory of M. Masuoka and H.-J.
Schneider by showing that there is a bijective correspondence between right
ideals coideals and right coideal subalgebras of any finite dimensional Hopf
algebra and we reformulate the still open problem in the general (i.e. infinite
dimensional) case. For cleft extensions we characterise closed elements of the
Galois connection as Hopf-Galois extensions. We describe the relation of our
results to the work of F. van Oystaeyen, Y. Zhang and also to the results of P.
Schauenburg on biGalois extensions.
