This is a review of recent results concerning the computation, from first principles, of correlation functions of quantum integrable systems. On the example of the XXZ Heisenberg chain, we will recall how to obtain, in the algebraic Bethe Ansatz framework, exact representations for the form factors and the two-point correlation functions in the finite chain. We will then explain how to analyze these expressions so as to derive, in the thermodynamic limit and in the massless regime of the chain, the long-distance asymptotic behavior of these two-point functions. We will conclude by an overview of some open problems and works in progress.