  We study a deformation of infinitesimal diffeomorphisms of a smooth manifold.
The deformation is based on a general twist. This leads to a differential
geometry on a noncommutative algebra of functions whose product is a
star-product. The class of noncommutative spaces studied is very rich.
Non-anticommutative superspaces are also briefly considered.
  The differential geometry developed is covariant under deformed
diffeomorphisms and it is coordinate independent. The main target of this work
is the construction of Einstein's equations for gravity on noncommutative
manifolds.
