  Given a Hopf algebra H, we study modules and bimodules over an algebra A that
carry an H-action, as well as their morphisms and connections. Bimodules
naturally arise when considering noncommutative analogues of tensor bundles.
For quasitriangular Hopf algebras and bimodules with an extra
quasi-commutativity property we induce connections on the tensor product over A
of two bimodules from connections on the individual bimodules. This
construction applies to arbitrary connections, i.e. not necessarily
H-equivariant ones, and further extends to the tensor algebra generated by a
bimodule and its dual. Examples of these noncommutative structures arise in
deformation quantization via Drinfeld twists of the commutative differential
geometry of a smooth manifold, where the Hopf algebra H is the universal
enveloping algebra of vector fields (or a finitely generated Hopf subalgebra).
  We extend the Drinfeld twist deformation theory of modules and algebras to
morphisms and connections that are not necessarily H-equivariant. The theory
canonically lifts to the tensor product structure.
