Abstract: Mild pro-p groups are finitely presented pro-p groups that, while not free, have a relation structure that is "free in the maximal possible way" in a certain precise sense. They play an important role in algebraic number theory, infinite Galois theory, and in the structure theory of pro-p groups, featuring, for instance, in the Gildenhuys–Serre conjecture and in Golod–Shafarevich theory. In this talk, we will explain the notion of mildness, present some of its key applications, and give a new cohomological criterion for a pro-p group to be mild.