Many of the most important results in mathematics are based on some inequality, of geometric or analytic nature. On the other hand, this separation between geometry and analysis is not sharp and the most intriguing inequalities are indeed the ones that have a mixed nature and enhance the interplay of the two realms. Moreover, many apparently purely geometric inequalities have some powerful functional counterpart, like for instance the Isoperimetric Inequality and Sobolev Inequality. I will try to give some general overview on geometric-analytic inequalities and will concentrate on one of them, precisely the Brunn-Minkowski inequality, an apparently geometric inequality which is at the core of modern convex geometry, and on its functional counterpart, the Borell-Brascamp-Lieb inequality. And also possibly show some applications to PDEs.