The project mixes two themes of Algebraic Geometry: the problem of extending and lifting torsors and the study of vector bundles as the main tool in the theory of the fundamental group scheme. Our two Research Packages have originally been inspired by each of them although the purpose of the project is precisely their interaction: the problems, the proposed approaches, and the participants have a background in either Arithmetic or Algebraic Geometry. Research Package 1 «Torsors » aims at studying the problem of extending and lifting of torsors. We are given a scheme X defined over a discrete valuation ring R and we consider its generic fiber XK over K (where K is the field of fractions of R). Let G be a finite K-group scheme and Y a G-torsor over XK. Is it possible to find a finite and flat R-group scheme G’, model of G, and a G’-torsor Y’ over X model of the given torsor Y over XK? This problem originally comes from Grothendieck’s celebrated Théorie de la spécialisation du groupe fundamental where the author of SGA1 proves that, possibly after extending scalars, the problem has a solution when R is a complete discrete valuation ring with algebraically closed residue field of positive characteristic p, with X proper and smooth over R with geometrically integral fibres where p does not divide the order of G. In the last decades there have been many improvements. Simmetrically let us consider the special fiber Xs of X over k=R/m, the residue field of R. Let H be a finite k-group scheme and Y a H-torsor over Xs. Is it possible to find a finite and flat R-group scheme H’, which lifts H, and a H’-torsor Y’ over X which lifts the given torsor Y over Xs? On this direction Pop recently proved the famous Oort conjecture but much more can still be studied. We will push to its limits a new approach using the deformation of essentially finite vector bundles. Research Package 2 « Vector Bundles and the Fundamental Group Scheme » concerns principally the study of the properties of the fundamental group scheme p(X,x) of a scheme X over a field k at a k-rational point x. Conjectured by Grothendieck and first defined by Nori it is, when X is reduced connected and proper over k, the k-group scheme naturally associated to the neutral tannakian category EF(X) whose objects are essentially finite vector bundles. Alternatively it can be constructed as the projective limit of finite and flat group schemes occurring as group of torsors over X. Thus in particular when k is just an algebraically closed field of characteristic zero it is nothing else but the étale fundamental group p^ét (X,x). Properties of the fundamental group schemes are in general difficult to study because for example it does not behaves well after base change; despite its complication it still shares some properties with the étale fundamental group. That is why it should hide inside itself many properties of the scheme and it is a natural object to study in the anabelian philosophy. The fundamental group scheme has been generalized in many different directions (by Gasbarri, Langer, Borne and Vistoli among others) and its properties have been widely studied for example by Mehta, Subramanian, Biswas, Esnault, Pauly and Antei. In particular let us recall that Gasbarri gave a construction for p(X,x) in the case of a scheme X over a Dedekind scheme S pointed at a S-valued point x. This makes clear how a deep knowledge of properties of the fundamental group scheme gives a usefool tool in answering to the main problem described in the section Research Package 1, and, of course, vice-versa. The project is coordinated by Marco Antei.