Summary: The steady extrusion of a Newtonian liquid through an annular die and its development outside and away from the die are studied under the influence of gravitational and surface tension forces. The finite element method is used for the simulations. The positions of the inner and outer free surface profiles are calculated simultaneously with other unknown fields, by using a Newton-Raphson iterative scheme. The effects of three relevant parameters, i.e. the Reynolds, the Stokes and the capillary numbers, on the shape of the annular film are studied for two values of inner to outer diameter ratio, corresponding to a thick and a thin annular film respectively. A one-dimensional model for the extrudate region, valid for thin annular films, is also presented, and its predictions are compared with two-dimensional finite element calculations. Despite the fact that it is valid away from the die exit, the one-dimensional model predicts satisfactorily the effects of Stokes and capillary numbers.