The celebrated Beurling-Ahlfors extension result characterizes the self-homeomorphisms of the unit circle which admit a quasiconformal extension to the disk. A remarkable feature of quasiconformal mappings is that their inverse map is also quasiconformal, implying further that both the map and its inverse have finite conformal energy. Recently with Onninen we investigated which boundary maps of the circle admit an extension to the disk satisfying the latter condition of having finite bi-conformal energy. In this talk we will address this problem and describe the extension method that provides a full characterization of the question for all Sobolev spaces. If time permits, I will also talk about some developments on other Sobolev mapping- and extension problems.