  Quantization of coordinates leads to the non-commutative product of
deformation quantization, but is also at the roots of string theory, for which
space-time coordinates become the dynamical fields of a two-dimensional
conformal quantum field theory. Appositely, open string diagrams provided the
inspiration for Kontsevich's solution of the long-standing problem of
quantization of Poisson geometry by virtue of his formality theorem. In the
context of D-brane physics non-commutativity is not limited, however, to the
topolocial sector. We show that non-commutative effective actions still make
sense when associativity is lost and establish a generalized
Connes-Flato-Sternheimer condition through second order in a derivative
expansion. The measure in general curved backgrounds is naturally provided by
the Born--Infeld action and reduces to the symplectic measure in the
topological limit, but remains non-singular even for degenerate Poisson
structures. Analogous superspace deformations by RR--fields are also discussed.
