Abstract: Campana orbifolds are pairs (X, Δ) that interpolate between the world of compact geometry and of logarithmic geometry. They come equipped with a sheaf of orbifold differential forms that generalises the ordinary cotangent sheaf and the logarithmic cotangent sheaf. After introducing these objects, I will turn my attention to the quest for positivity. In particular the quest for finding projective varieties with ample cotangent bundles; I will highlight some recent work by Brotbeck–Deng and Brotbeck–Darondeau in providing examples of varieties with ample cotangent bundles and positive logarithmic cotangent bundles. Finally I would like to say something about positivity of Campana orbifolds.