Abstract
We describe the Minimal Model Program in the family of $\mathbb{Q}$-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP in toric varieties due to M.~Reid, and we complete the results on MMP in spherical varieties due to M.~Brion in the case of horospherical varieties.