Incarnations of Berthelot's conjecture

In this article we give a survey of the various forms of Berthelot's conjecture and some of the implications between them. By proving some comparison results between push-forwards of overconvergent isocrystals and those of arithmetic DD-modules, we manage to deduce some cases of the conjecture from Caro's results on the stability of overcoherence under push-forward via a smooth and proper morphism of varieties. In particular, we show that Ogus' convergent push-forward of an overconvergent F-isocrystal under a smooth and projective morphism is overconvergent.