The Brunn-Minkowski inequality is a fundamental result in convex geometry controlling the volume of  the sum of subsets of ℝn. It asserts that for  sets A,B⊂ℝn of equal volume and a parameter t∈(0,1), we have |tA+(1−t)B|≥|A| with equality iff A=B is convex.  Early work of Ruzsa, as well as results on the special case A=B suggested a linear stability result; if |tA+(1−t)B|≤(1+δ)|A|, then |co(A)∖A|=On,t(δ)|A|. On the other hand, the study of the stability of the isoperimetric inequality supported a folklore quadratic stability conjecture; for δ as before, we have (up to translation) |A△B|=On,t(δ√)|A|. In this talk we explore the background of these conjectures, their connection with the discrete context of additive combinatorics (in particular the connection with geometric instances of the Polynomial Freiman-Ruzsa conjecture), and the proof of the linear conjecture. Joint work with Alessio Figalli and Marius Tiba.