  We work out the general features of perturbative field theory on
noncommutative manifolds defined by isospectral deformation. These (in general
curved) `quantum spaces', generalizing Moyal planes and noncommutative tori,
are constructed using Rieffel's theory of deformation quantization for action
of $\R^l$. Our framework, incorporating background field methods and tools of
QFT in curved spaces, allows to deal both with compact and non-compact spaces,
as well as with periodic or not deformations, essentially in the same way. We
compute the quantum effective action up to one loop for a scalar theory,
showing the different UV/IR mixing phenomena for different kinds of isospectral
deformations. The presence and behavior of the non-planar parts of the Green
functions is understood simply in terms of off-diagonal heat kernel
contributions. For periodic deformations, a Diophantine condition on the
noncommutativity parameters is found to play a role in the analytical nature of
the non-planar part of the one-loop reduced effective action. Existence of
fixed points for the action may give rise to a new kind of UV/IR mixing.
