  We study a self-interacting scalar $\varphi^4$ theory on the $d$-dimensional
noncommutative torus. We determine, for the particular cases $d=2$ and $d=4$,
the counterterms required by one-loop renormalization. We discuss higher loops
in two dimensions and two-loop contributions to the self-energy in four
dimensions. Our analysis points toward the absence of any problems related to
the UV/IR mixing and thus to renormalizability of the theory. However, we find
another potentially troubling phenomenon which is a wild behavior of the
two-point amplitude as a function of the noncommutativity matrix $\theta$.
