Abstract: There has been much progress recently in proving single-letter formulas for the mutual information (or " free energy ") in high-dimensional estimation and learning problems. Computing the mutual information is important in order to locate the various " phase transitions " occurring in such problems when the noise increases or the data becomes too scarce. It is also key in computing various optimal achievable errors. Unfortunately all existing methods are highly involved, difficult to generalize and restricted in their applicability. In this talk I'll present a new method, called " adaptive interpolation method ", that eliminates these barriers all at once: It is much simpler, very generic and able to tackle problems that were resisting until now. I will illustrate the method on a paradigmatic model of high-dimensional estimation, namely the " Wigner spiked model " (or " low-rank matrix factorization "). I will also briefly review some models that are now under full rigorous control thanks to this approach, as well as very recent extensions to physics models such as the " ferromagnetic p-spin model on sparse random graphs ", an open problem for decades for reasons that I'll mention.