  A horospherical variety is a normal algebraic variety where a reductive
algebraic group acts with an open orbit which is a torus bundle over a flag
variety. For example, toric varieties and flag varieties are horospherical. In
this paper, we classify Fano horospherical varieties in terms of certain
rational polytopes that generalize the reflexive polytopes considered by V.
Batyrev. Then, we obtain an upper bound on the degree of smooth Fano
horospherical varieties, analogus to that given by O. Debarre in the toric
case. We extend a recent result of C. Casagrande: the Picard number of any Fano
Q-factorial horospherical variety is bounded by twice the dimension.
