Despite many attempts, quantum field theory on commutative space-time has severe shortcomings. Recently various deformations of the algebra of functions over a manifold have been considered. We review such regularization methods based on matrix geometry and show, how an ultraviolet cut-off respecting symmetries results. Nontrivial topological configurations as well as supersymmetric models can be treated as well. Next we review recent attempts to renormalize deformed quantum field theory models. We mention the IR-UV mixing which is avoided if a higher symmetry is involved. Whether the Yang-Mills model is renormalizable is still open.
| E-Mail: | harald.grosse@univie.ac.at |