  We study the notion of a Dirac operator in the framework of twist-deformed
noncommutative geometry. We provide a number of well-motivated candidate
constructions and propose a minimal set of axioms that a noncommutative Dirac
operator should satisfy. These criteria turn out to be restrictive, but they do
not fix a unique construction: two of our operators generally satisfy the
axioms, and we provide an explicit example where they are inequivalent. For
highly symmetric spacetimes with Drinfeld twists constructed from sufficiently
many Killing vector fields, all of our operators coincide. For general
noncommutative curved spacetimes we find that demanding formal self-adjointness
as an additional condition singles out a preferred choice among our candidates.
Based on this noncommutative Dirac operator we construct a quantum field theory
of Dirac fields. In the last part we study noncommutative Dirac operators on
deformed Minkowski and AdS spacetimes as explicit examples.
