  We establish the vanishing of the third unramified cohomology group for many
types of Fano hypersurfaces in projective space over an algebraically closed
field of arbitrary characteristic, and over a finite field. For cubic
hypersurfaces over a finite field, the case of fourfolds remains open.
  ---
  Sur un corps alg\'ebriquement clos et sur un corps fini, on \'etablit de
nouveaux r\'esultats d'annulation pour la cohomologie non ramifi\'ee de degr\'e
3 pour de nombreux types d'hypersurfaces de Fano. Le cas des hypersurfaces
cubiques de dimension 4 sur un corps fini reste ouvert.
