The isoperimetric inequality asserts that, given an open set with a fixed volume, the ball is the unique minimizer of its perimeter. The question of stability arises: when the perimeter of the set closely approaches the perimeter of a ball, does the set closely resemble a ball? I will sketch a proof of the sharp quantitative version of this statement due to Cicalese and Leonardi. The novelty of their result is their ingenious quantitative compactness argument which reduces the problem to studying smooth deformations of the ball.