  We look in Euclidean $R^4$ for associative star products realizing the
commutation relation $[x^\mu,x^\nu]=i\Theta^{\mu\nu}(x)$, where the
noncommutativity parameters $\Theta^{\mu\nu}$ depend on the position
coordinates $x$. We do this by adopting Rieffel's deformation theory
(originally formulated for constant $\Theta$ and which includes the Moyal
product as a particular case) and find that, for a topology $R^2 \times R^2$,
there is only one class of such products which are associative. It corresponds
to a noncommutativity matrix whose canonical form has components
$\Theta^{12}=-\Theta^{21}=0$ and $\Theta^{34}=-\Theta^{43}= \theta(x^1,x^2)$,
with $\th(x^1,x^2)$ an arbitrary positive smooth bounded function. In Minkowski
space-time, this describes a position-dependent space-like or magnetic
noncommutativity. We show how to generalize our construction to $n\geq 3$
arbitrary dimensions and use it to find traveling noncommutative lumps
generalizing noncommutative solitons discussed in the literature. Next we
consider Euclidean $\lambda\phi^4$ field theory on such a noncommutative
background. Using a zeta-like regulator, the covariant perturbation method and
working in configuration space, we explicitly compute the UV singularities. We
find that, while the two-point UV divergences are non-local, the four-point UV
divergences are local, in accordance with recent results for constant $\Theta$.
