I describe the results of a long term project on the
calculation and clarification of the mathematical structure of the
correlation functions of integrable systems. Our main model system
in this context is the XXZ spin chain which includes several interesting
relativistic and non-relativistic 1+1d quantum field theories as scaling
limits and also appears as a model for highly anisotropic magnetic
materials in solid state physics. Recently, it was established that
the static correlation functions of the XXZ-chain all factorize
(T. Miwa, M. Jimbo and F. Smirnov, arXiv:0811.0439) under rather general
conditions. This means that all correlation functions can be expressed
in terms of the one-point function and a special neighbour two-point
function (which may be understood as a kind of Wick theorem for an
interacting system). Those functions admit an efficient description
by means of linear and non-linear integral equation (H. Boos and F.
G\"ohmann, arXiv:0903.5043) which, in turn allows us to calculate
short-range correlation functions for the spin chain, say, at finite
temperature, in the thermodynamic limit with arbitrary numerical
precision (e.g. C. Trippe, F. G\"ohmann and A. Kl\"umper arXiv:0908.2232
and arXiv:0912.1739).
