  We review our recent construction of the $\phi^4$-model on four-dimensional
Moyal space. A milestone is the exact solution of the quartic matrix model
$Z[E,J]=\int d\Phi \exp(tr(J\Phi- E\Phi^2 -(\lambda/4) \Phi^4))$ in terms of
the solution of a non-linear equation for the 2-point function and the
eigenvalues of $E$. The $\beta$-function vanishes identically. For the Moyal
model, the theory of Carleman type singular integral equations reduces the
construction to a fixed point problem. Its numerical solution reveals a
second-order phase transition at $\lambda_c\approx-0.396$ and a phase
transition of infinite order at $\lambda=0$. The resulting Schwinger functions
in position space are symmetric and invariant under the full Euclidean group.
They are only sensitive to diagonal matrix correlation functions, and
clustering is violated. The Schwinger 2-point function is reflection positive
iff the diagonal matrix 2-point function is a Stieltjes function. Numerically
this seems to be the case for coupling constants $\lambda \in [\lambda_c,0]$.
