We had three main themes on the relation between the fundamental group of varieties, notably over finite fields, and various types of isocrystals: Gieseker’s conjecture: vanishing of the étale fundamental group should force the infinitesimal crystals to be trivial. Recall we prove the conjecture on smooth projective varieties over an algebraically closed field of characteristic p > 0 with Mehta, this was the starting point of the proposal. We progressed with V. Srinivas (Tata Institute of Fundamental Research, Mumbai) proving a version of Gieseker’s conjecture on singular projective varieties over finite fields and a relative version. de Jong’s conjecture: vanishing of the étale fundamental group on smooth projective varieties over an algebraically closed field of characteristic p > 0 should force the isocrystals to be trivial. This is a very profound conjecture and we do not have yet a complete answer. Yet with Atsushi Shiho (Tokyo University) we could prove it under some restriction on the geometry of the variety (Annales de l’Insititut Fourier). On the way we proved a vanishing theorem on the crystalline Chern class of locally free (or convergent) isocrystals. This is the starting point of new studies, notably of Bhatt-Lurie. Deligne’s conjecture: on a smooth variety over a finite field we could prove, with Tomoyuki Abe (KAVLI, Tokyo University) the existence of ℓ-adic companions to overconvergent F-isocrystals. The method and the result were already used in a number of applications. Moreover Kedlaya proved afterwards the same theorem a using different method. Simpson’s conjecture: rigid complex local systems on smooth projective varieties are integral. We could use in a first proof isocrystals and their ℓ-adic companions to prove it with Michael Groechenig, and found later on a shorter proof solely based on Drinfeld’s ℓ-adic a companions. The result has a number of applications, e.g. in Köhler geometry.