Symmetries, covariant derivatives and gravity on noncommutative spacetime

Noncommutative spacetimes and their symmetries can be constructed using the notion of (abelian) Drinfeld twist. We review the general formalism and present the examples of the Poincare *-Lie algebra on Moyal-Weyl deformed Minkowski space and of the *-Lie algebra of vectorfields on an arbitrary twist deformed manifold. The second example defines the notion of Lie derivative. This paves the way to the definition of the covariant derivative. Noncommutative Einstein's gravity equations are formulated.