  It has been proposed that the noncommutative geometry of the "fuzzy" 2-sphere
provides a nonperturbative regularization of scalar field theories. This
generalizes to compact Kaehler manifolds where simple field theories are
regularized by the geometric quantization of the manifold.
  In order to permit actual calculations and the comparison with other
regularizations, I describe the perturbation theory of these regularized models
and propose an approximation technique for evaluation of the Feynman diagrams.
I present example calculations of the simplest diagrams for the $\phi^4$ model
on the 2-sphere, a 2-sphere times a 2-sphere, and 2-dimensional complex
projective space.
  This regularization fails for noncompact spaces; I give a brief dimensional
analysis argument as to why this is so. I also discuss the relevance of the
topology of Feynman diagrams to their ultra-violet and infra-red divergence
behavior in this model.
