  This paper is a written form of a talk. It gives a review of various notions
of Galois (and in particular cleft) extensions. Extensions by
coalgebras,bialgebras and Hopf algebras (over a commutative base ring) and by
corings,bialgebroids and Hopf algebroids (over a non-commutative base algebra)
are systematically recalled and compared.
  In the first version of this paper, the journal version of [15, Theorem 2.6]
was heavily used, in two respects. First, it was applied to establish an
isomorphism between the comodule categories of two constituent bialgebroids in
a Hopf algebroid. Second, it was used to construct a Morita context for any
bicomodule for a coring extension. Regrettably, it turned out that the proof of
[15, Theorem 2.6] contains an unjustified step. Therefore, our derived results
are not expected to hold at the stated level of generality either. In the
revised version we make the necessary corrections in both respects. In doing
so, we obtain a corrected version of \cite[5, Theorem 4.2] as well, whose
original proof contains a very similar error to [15, Theorem 2.6].
