Which QFTs were rigorously constructed? 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.
 A: I assume you know that free field theories can be constructed (in arbitrary dimension of spacetime, I believe).
In algebraic quantum field theory (a la Haag), there is for example the conformal Ising model. You can find more about this in these references:


*

*Mack, G., & Schomerus, V. (1990). Conformal field algebras with quantum symmetry from the theory of superselection sectors. Communications in  Mathematical Physics, 134(1), 139–196.

*Böckenhauer, J. (1996). Localized endomorphisms of the chiral Ising model. Communications in  Mathematical Physics, 177(2), 265–304.


In the latter "localized endomorphisms" as in the Doplicher-Haag-Roberts programme on superselection sectors. See for example this paper on the arXiv.
There are surely more examples, also in the Wightman setting, but I'm not too familiar with them.
A: An approach to the rigorous construction of gauge theories is via the lattice.  There were some papers in the 1980s -- I remember those of Tadeusz Bałaban (MathSciNet) (inSPIRE) in Communications -- on this topic.
A: All QFT's on lattice are well defined. It may be true that
all well defined QFT's are either lattice theory or the low energy limit
of a lattice theory. See a related post Rigor in quantum field theory
A: The list would be a bit too long here. It also depends on how demanding you are on the notion
of "being constructed". If you take a rather restrictive definition as: all the Wightman axioms have been established then that excludes Yang-Mills even though important work has been done by Bałaban as mentioned by José and also other authors: Federbush, Magnen, Rivasseau, Sénéor.
Examples of theories where all the Wightman axioms have been checked:

*

*Massive 2d scalar theories with polynomial interactions, see this article by Glimm, Jaffe and Spencer.


*Massive $\phi^4$ in 3d, see this article by Feldman and Osterwalder
as well as this one by Magnen and Sénéor.


*Massive Gross-Neveu in 2d see this article by Gawędzki and Kupiainen and this one by Feldman, Magnen, Rivasseau and Sénéor.


*Massive Thirring model, see this article by Fröhlich and Seiler and this more recent one by Benfatto, Falco and Mastropietro.
A: For CFT there are many examples. I will give some examples of local conformal nets on the circle (or real line).
The Ising model Pieter mentions is the Virasoro net with $c=1/2$. The Virasoro net can be constructed for the discrete $c<1$ and $c>1$. See eg.

*

*Kawahigashi Y.  Longo R. (2004) "Classification of local conformal
nets. Case c<1" Ann. of Math. 160, p493-522

They furthermore classify all local conformal nets with central charge $c<1$.
Positive energy representations of loop groups give conformal nets.

*

*Jürg Fröhlich and Fabrizio Gabbiani. Operator algebras and conformal field theory. Comm. Math. Phys. Volume 155, Number 3 (1993), 569-640. Link
The conformal nets associated to lattices and its orbifolds are constructed in

*

*Dong & Xu. Conformal nets associated with lattices and their orbifolds. Advances in Mathematics (2006)
Volume: 206, Issue: 1, Pages: 279-306

and in the same issue Kawahigashi and Longo have constructed the "moonshine" net.

*

*Kawahigashi & Longo. Local conformal nets arising from framed vertex operator algebras. Adv. Math. 206 (2006), 729-751.

For massive models in 2D Lechner constructed the factorizing S-matrix models in which are a priori just "wedge-local" nets but he managed to show for a class that to show the existence of local observables.

*

*Lechner. Construction of Quantum Field Theories with Factorizing S-Matrices. Commun.Math.Phys. 277, 821-860 (2008)

A: Notice that a conformal AQFT net as in the replies of Marcel and Pieter only gives the "chiral data" of a CFT, not a full CFT defined on all genera. For the rational case the full 2d CFTs have been constructed and classified by FFRS. Also Liang Kong has developed notions that promote a chiral CFT to a full CFT (rigorously), see this review.
Beyond that, of course topological QFTs have been rigorously constructed, including topological sigma-models on nontrivial targets. Via "TCFT" this includes the A-model and the B-model in 2d. 
