Philippe Gille University of Lyon, France, and IMAR, Bucharest, Romania This is a report on joint work with Ofer Gabber and Laurent Moret-Bailly. Let K be the fraction field of a henselian valuation ring R of positive characteristic p. Let Y be a K-variety, H an algebraic group over K, and f:X→Y an H-torsor over Y. We consider the induced map X(K)→Y(K) which is continuous for the topologies coming from the valuation. If I denotes the image of this map, we investigate the following questions: (a) Is I locally closed (resp., closed) in Y(K)? (b) Is the continuous bijection X(K)/H(K)→I a homeomorphism?