The unsteady extrusion of a viscoelastic film from an annular and axisymmetric die is examined. This moving boundary problem is solved by mapping the inner and outer liquid/air interfaces of the extruded film onto fixed ones, and by transforming the governing equations accordingly. The ratio of the film thickness to its inner radius at the exit of the die is used as the small parameter, epsilon, in a regular perturbation expansion of the governing equations. Forces applied on the film give rise to four dimensionless numbers: Stokes, Capillary, Reynolds and Deborah. When the Oldroyd-B model is used, the dimensionless retardation time also arises. For typical fluid properties and process conditions, the Stokes and Deborah numbers are O(epsilon(0)), i.e. much larger than the other relevant dimensionless numbers. In such cases, the base state is significantly deforming with time and it is calculated numerically by solving a partial differential system of equations in time and the axial direction. Special attention is required for its accurate numerical solution. It was found that gravity plays the most important role in the process by accelerating the film, deflecting its inner and outer surfaces towards its axis of symmetry and decreasing its thickness around the middle of its length. For typical values of the De number, its increase leads to deceleration of the film that has less curved interfaces and more uniform thickness along its length. These effects become apparent, if the St number is of order one; if it is smaller, the effects of fluid elasticity decrease considerably. For typical values of the Ca and Re numbers, and of the retardation time of the fluid, their influence on the process is small. (C) 2000 Elsevier Science B.V. All rights reserved.