The classical isoperimetric inequality prescribes the maximal volume of a body in $\mathbb{R}^n$ under a perimeter constraint. Since the optimum is given by the ball $B$, it gives $P(E)\geq n|E|^{1-\frac 1n}|B|^{\frac 1n}$ for any $E\subset\mathbb{R}^n$. Its anisotropic version concerns the $K-$perimeter $P_K$, defined from a convex body $K\subset\mathbb{R}^n$, and it reads $P_K(E)\geq n|E|^{1-\frac 1n}|K|^{\frac 1n}$; the optimum is realized by $K$. A quantitative version of these inequalities means estimating the difference $P(E)\!-\!n|E|^{1-\frac 1n}|B|^{\frac 1n}$ in terms of ``how much $E$ is different from $B$''. The optimal quantitative version of the classical inequality has been proved in 2008 by Fusco, Maggi and Pratelli through some symmetrization methods, peculiar to the isotropic case. The works that I'll present have allowed, thanks to the application of the Brenier transport, to get the same result in the anisotropic case.