  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.
