We consider a Hopf Galois extension B ⊂ A, for A a comodule algebra over the Hopf
algebra H with coinvariant algebra B. After giving a number of examples, we discuss
Galois extensions with additional properties, such as having a normal basis. We then
consider when there is a category equivalence between the category of modules over
B and the category of “relative Hopf modules” for A and H. Finally we discuss more
recent work of van Oystaeyen and Zhang and of Schauenburg on obtaining
correspondence theorems between suitable subalgebras of A and Hopf ideals of
H.