  In this memoir, we consider Hopf Galois extensions and families of
noncommutative rings known as skew polynomial rings, PBW extensions and skew
PBW extensions. We address remarkable examples of Hopf Galois extensions (e.g.,
Hopf algebras, field extensions, strongly graded algebras, crossed products,
principal bundles), some properties and recent advances in the theory such as
quantum torsors and Hopf Galois systems. Next, we study the families of
noncommutative rings and some of its basic properties and recall remarkable
examples appearing in the theory. Finally, some interactions between all
families of rings are presented.
