In this paper we study the question of when anH-comodule algebra is a faithfully flat Galois extension of its sub algebra of coinvariants for certain Hopf algebrasH. We note that ifHis connected, a faithfully flat Galois extension must actually be cleft and hence a crossed product, and we show that with a different hypothesis, a faithfully flat Galois extension must be a smash product. We also describe faithfully flat Galois extensions whenHis pointed cocommutative. We give an explicit description ofH-comodule algebras whenHis a polynomial algebra, a divided power Hopf algebra, a free algebra, or a shuffle algebra. We give necessary and sufficient conditions for anH-extension to be faithfully flat Galois in these cases and in the case whereHis the enveloping algebra of a Lie algebra; a key ingredient in our analysis is the existence and description of a total integral. In the case where, we give a simple example of a flat Galois extension that is not faithfully flat.