Recently the mathematical structure of the static correlation functions of the spin-1/2 Heisenberg chain has been completely understood. It has been proved by Boos, Jimbo, Miwa and Smirnov that, after appropriate regularizations, all static correlation functions become polynomials in a one-point and a special neighbour-two-point correlation function. We review this important result and show that those two functions allow for an efficient description in terms of the solutions of linear and non-linear integral equations. This opens the way for applications in quantum field theory and condensed matter physics.