Final Activity Report Summary - NC FUZZY GAUGE FIELD (Simulating Chiral Gauge Theory with Matrix Models : Noncommutative Geometry and Fuzzy Approximations) Field theory in a space-time with non-commuting coordinates is a promising approach to understand the physics at smallest, yet still not accessible, scales. The investigation of the properties of such theories and models is becoming a more and more exciting topic for field theoreticians working on models beyond the standard ones of elementary particles. Since non-perturbative methods are also required, Monte Carlo techniques, well-known in statistical physics, lattice gauge theories and matrix models, can be successfully applied. The aim of this project was to perform theoretical and numerical investigations of the phase structure of the two-dimensional non-commutative fuzzy Yang-Mills theory and of the non-commutative fuzzy Schwinger model. During the two-year period this plan became naturally extended to obtain emergent geometry from matrix models and simulate supersymmetry. The last point was the most exciting one, since it could lead to the full simulation of M-theory. The discussion of the non-cummutative Schwinger model might also be relevant for supersymmetry, since it involved the simulation of fermions on non-commutative spaces. It turned out to be additionally interesting from a quantum chromodynamics point of view.