This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien-Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the M model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated. (c) 2007 Elsevier B.V. All rights reserved.