TITLE: Linear/Non-linear sigma model
QUESTION [13 upvotes]: This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model. 
I gauge from some other interactions that it has considerable interest even in mathematics (geometry?) apart from QFT. 
I would like if people can type in what are the canonical explanatory/expository references about the linear and non-linear sigma model from both the QFT and mathematical points of view. It would be illuminating if experts can shed light on why is this model exciting again from both the QFT and mathematics point of view. 
{As an example of the kind of expository reference I am looking for I would like to mention this one as an example for another interesting model in QFT}

REPLY [3 votes]: It is strange to distinguish the QFT point of view from the mathematics point of view. That's comparing apples to oranges.
One can distinguish the physics (non-mathematically rigorous) point of view from that of rigorous mathematics. In this case QFT has overlaps with both. I think the distinction here is between classical sigma models, an example of geometric PDE, and quantum sigma models which involves probability on a space of fields. As Willie Wong covered classical sigma models from the mathematics point of view and José covered quantum sigma models from the physics point of view; let me add some pointers to the literature regarding quantum sigma models from the mathematics point of view.
For rigorous work on perturbative (in the sense of formal power series) renormalization of the quantum sigma model, see:

P. K. Mitter and T. R. Ramadas,
"The two-dimensional $O(N)$ nonlinear $\sigma$-model: renormalisation and effective actions", Comm. Math. Phys. 122 (1989), no. 4, 575–596. 
T. Nguyen,
"Quantization of the nonlinear sigma model revisited",
J. Math. Phys. 57 (2016), no. 8, 082301, 40 pp. 

For rigorous nonperturbative results see:

C. Kopper,
"Mass generation in the large N-nonlinear σ-model",
Comm. Math. Phys. 202 (1999), no. 1, 89–126. This proves a mass gap but in the presence of an ultraviolet cutoff. 
K. Gawędzki and A. Kupiainen,
"Continuum limit of the hierarchical $O(N)$ nonlinear $\sigma$-model".
Comm. Math. Phys. 106 (1986), no. 4, 533–550. This removes the ultraviolet cutoff in a finite volume but concerns a hierarchical toy model.

Also, another account from the physics point of view and related to José's answer, the Ricci flow and all that, is the article:

M. Carfora, "Renormalization group and the Ricci flow",
Milan J. Math. 78 (2010), no. 1, 319–353.