Which QFTs have mathematically rigorous constructions a la AQFT? I understand there are many such constructions in 2D, in particular 2D CFT has been extensively studied mathematically. But even in 2D there are many theories without known constructions e.g. non-linear sigma models in most curved target spaces. In higher dimensions the list of non-free examples is much shorter.
I'm looking for a complete list of QFTs constructed to-date with reference to each construction. Also, a good up-to-date review article of the entire subject would be nice.
EDIT: This question concerns QFTs in Minkowski (or at least Euclidean) spacetime, not spacetimes with curvature and/or non-trivial topology.