Abstract: Galois theory for Hopf algebras has its roots in works of Chase, Harrison and Rosenberg in 1965 where they wanted to extend the classical Galois theory of fields. Fifteen years later, Kreimer and Takeuchi defined Hopf Galois extensions which are noncommutative analogue of torsors and principal bundles.

In this talk we explain the basics of Hopf Galois extensions and introduce several examples. Furthermore we explain the homology theories related to a Hopf Galois extension and explain some new results.