  We study classical scalar field theories on noncommutative curved spacetimes.
Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005),
3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative
spacetimes by using (Abelian) Drinfel'd twists and the associated *-products
and *-differential geometry. In particular, we allow for position dependent
noncommutativity and do not restrict ourselves to the Moyal-Weyl deformation.
We construct action functionals for real scalar fields on noncommutative curved
spacetimes, and derive the corresponding deformed wave equations. We provide
explicit examples of deformed Klein-Gordon operators for noncommutative
Minkowski, de Sitter, Schwarzschild and Randall-Sundrum spacetimes, which solve
the noncommutative Einstein equations. We study the construction of deformed
Green's functions and provide a diagrammatic approach for their perturbative
calculation. The leading noncommutative corrections to the Green's functions
for our examples are derived.
