As a generalization of Mori's characterization of the projective space, Campana and Peternell conjectured that the only Fano manifolds with nef tangent bundles are rational homogeneous.
In this talk, we review some properties of Fano manifolds with nef tangent bundles and some partial classification results. Furthermore, we propose a strategy to study the Campana-Peternell conjecture, based on my joint work with R. Munoz, G. Occhetta, L E. Sola Conde and J. Wisniewski.