  We investigate the phase structure of non-commutative scalar field theories
and find evidence for ordered phases which break translation invariance. A
self-consistent one-loop analysis indicates that the transition into these
ordered phases is first order. The phase structure and the existence of scaling
limits provides an alternative to the structure of counter-terms in determining
the renormalizability of non-commutative field theories. On the basis of the
existence of a critical point in the closely related planar theory, we argue
that there are renormalizable interacting non-commutative scalar field theories
in dimensions two and above. We exhibit this renormalization explicitly in the
large $N$ limit of a non-commutative O(N) vector model.
