in this talk i will present a new method for studying the regularity of minimizers of some variational problems, including in particular some classical free-boundary problems. using as a model case the so-called obstacle problem, i will explain what regularity of the free-boundary means and how we obtain it by using a new tool, called (log) -epiperimetric inequality. this technique is very general, and much like caffarelli's 'improvement of flatness' for regular points, it allows for a uniform treatment of singularities in many different free-boundary problems. moreover it is able to deal with logarithmic regularity, which in the case of the obstacle problem is optimal due to an example of figalli-serra. if time permits i will explain how such an inequality is linked to the behavior of a gradient flow at infinity.