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We study quartic matrix models with partition function Z[E,J]=\int dM\exp(trace(JM-EM^2-(\lambda/4)M^4)). The integral is over the space ofHermitean NxN-matrices, the external matrix E encodes the dynamics, \lambda>0is a scalar coupling constant and the matrix J is used to generate correlationfunctions. For E not a multiple of the identity matrix, we prove a universalalgebraic recursion formula which gives all higher correlation functions interms of the 2-point function and the distinct eigenvalues of E. The 2-pointfunction itself satisfies a closed non-linear equation which must be solvedcase by case for given E. These results imply that if the 2-point function of aquartic matrix model is renormalisable by mass and wavefunctionrenormalisation, then the entire model is renormalisable and has vanishing\beta-function.  As main application we prove that Euclidean \phi^4-quantum field theory onfour-dimensional Moyal space with harmonic propagation, taken at itsself-duality point and in the infinite volume limit, is exactly solvable andnon-trivial. This model is a quartic matrix model, where E has for N->\inftythe same spectrum as the Laplace operator in 4 dimensions. Using the theory ofsingular integral equations of Carleman type we compute (for N->\infty andafter renormalisation of E,\lambda) the free energy density(1/volume)\log(Z[E,J]/Z[E,0]) exactly in terms of the solution of a non-linearintegral equation. Existence of a solution is proved via the Schauder fixedpoint theorem.  The derivation of the non-linear integral equation relies on an assumptionwhich we verified numerically for coupling constants 0<\lambda\leq (1/\pi).

Self-dual noncommutative \phi^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory