  Field theories on deformed spaces suffer from the IR/UV mxing and
renormalization is generically spoiled. In work with R. Wulkenhaar, one of us
realized a way to cure this desease by adding one more marginal operator. We
review these ideas, show the application to $\phi^3$ models and use heat kernel
expansion methods for a scalar field theory coupled to an external gauge field
on a $\theta$-deformed space and derive noncommutative gauge actions.
