In 1991 Campana and Petemell proposed, as a natural
algebro-geometric extension of Mori’s characterization of the projective
space, the problem of classifying the complex projective Fano manifolds
whose tangent bundle is nef, conjecturing that the only varieties satisfy
ing these properties are rational homogeneous. In this paper we review
some background material related to this problem, with special attention
to the partial results recently obtained by the authors.