Generic positivity and applications to hyperbolicity of moduli spaces

The proof of the celebrated Viehweg's hyperbolicity conjecture is a consequence of two remarkable results: Viehweg and Zuo's existence results for global pluri-differential forms induced by variation in a family of canonically po-larised manifolds and Campana and P\v{a}un's vast generalisation of Miyaoka's generic semipositivity result for non-uniruled varieties to the context of pairs. The aim of this chapter is an exposition of Campana-P\v{a}un's generic semipositivity theorem .