We outline how Drinfeld twist deformation techniques can be applied to the
deformation quantization of principal bundles into noncommutative principal
bundles, and more in general to the deformation of Hopf-Galois extensions.
First we twist deform the structure group in a quantum group, and this leads to
a deformation of the fibers of the principal bundle. Next we twist deform a
subgroup of the group of authomorphisms of the principal bundle, and this leads
to a noncommutative base space. Considering both deformations we obtain
noncommutative principal bundles with noncommutative fiber and base space as
well.Comment: 20 pages. Contribution to the volume in memory of Professor Mauro
Francaviglia. Based on joint work with Pierre Bieliavsky, Chiara Pagani and
Alexander Schenke
