Stability in geometric inequalities

I will review some results on the stability of some geometric inequalities. The notion of stability considers the following question: Suppose that a function almost attains equality in some inequality for which minimizers are known. Can we prove that such a function is close to one of the minimizers? Over the past decade, the method of mass transportation made significant progress on this issue. In this talk, I will focus on the Brunn-Minkowski inequality and the isoperimetric inequality and give a quick overview of the proof strategy.