Low temperature asymptotic expansions in spin O(N) models in dimensions 3 and higher

I will review old and recent results on the problem of obtaining asymptotic series in powers of the temperature (T) around T=0 for local correlation functions of lattice spin models with continuous symmetries, using Reflection Positivity. After presenting the models and the problem, I will mostly discuss how to obtain the formal power series in a way that can be translated into a proof, and only briefly mention how to actually turn the formal procedure into a proof. If time permits, I will also mention some features of the d=2 case.
Based on recent joint work with A. Giuliani, and on (less recent) work of Bricmont, Fontaine, Lebowitz, Lieb, and Spencer.