The motion of a two-dimensional glacier is considered. At each time step, given the shape of the glacier, ice is modelled as an incompressible non-Newtonian fluid and a non-linear elliptic problem has to be solved to obtain the horizontal velocity field. Then, the upper surface of the glacier is updated by solving a transport equation. Finite element techniques are used to compute the velocity field whereas the transport equation is solved using a Lax-Wendroff scheme. Numerical results are compared to experiments on Gries glacier (Wallis, Switzerland) between 1961 and 1991. Then, a predition for 2021 is proposed. Copyright (C) 2004 John Wiley Sons, Ltd.