  We construct noncommutative principal bundles deforming principal bundles
with a Drinfeld twist (2-cocycle). If the twist is associated with the
structure group then we have a deformation of the fibers. If the twist is
associated with the automorphism group of the principal bundle, then we obtain
noncommutative deformations of the base space as well. Combining the two twist
deformations we obtain noncommutative principal bundles with both
noncommutative fibers and base space. More in general, the natural isomorphisms
proving the equivalence of a closed monoidal category of modules and its twist
related one are used to obtain new Hopf-Galois extensions as twists of
Hopf-Galois extensions. A sheaf approach is also considered, and examples
presented.
