  We study the IR/UV connection of the four-dimensional non-commutative phi^4
theory by using the Wilsonian Renormalization Group equation. Extending the
usual formulation to the non-commutative case we are able to prove UV
renormalizability to all orders in perturbation theory. The full RG equations
are finite in the IR, but perturbative approximations of them are plagued by IR
divergences. The latter can be systematically resummed in a way analogous to
what is done in finite temperature field theory. As an application,
next-to-leading order corrections to the two-point function are explicitly
computed. The usual Wilsonian picture, i.e. the insensitivity of the IR regime
to the UV, does not hold in the non-commutative case. Nevertheless it can be
partially recovered by a matching procedure, in which a high-energy theory,
defined in the deep non-commutative regime, is connected at some intermediate
scale to a commutative low-energy theory. The latter knows about
non-commutativity only through the boundary conditions for two would-be
irrelevant couplings.
