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