  In this contribution to the proceedings of the Corfu Summer Institute 2015, I
give an overview over quantum field theories on non-commutative Moyal space and
renormalization. In particular, I review the new features and challenges one
faces when constructing various scalar, fermionic and gauge field theories on
Moyal space, and especially how the UV/IR mixing problem was solved for certain
models. Finally, I outline more recent progress in constructing a
renormalizable gauge field model on non-commutative space, and how one might
attempt to prove renormalizability of such a model using a generalized
renormalization scheme adapted to the non-commutative (and hence non-local)
setting.

High Energy Physics - Theory;General Relativity and Quantum Cosmology