  In 1991 Campana and Peternell 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 satisfying 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.
