The Lattice Boltzmann Method (LBM) is a technique based on a discretisation of the Boltzmann equation. LBM for Newtonian fluids has been used as a computational fluid dynamics approach for nearly a decade and numerical experiments on complex flows, such as the suspension of thousands hard spheres or the calculation of transport coefficients in extremely complex porous media, have shown that LBM is quantitatively accurate and computationally efficient. We have recently proposed, Giraud et al . (1997), to use the Lattice Boltzmann method as an alternative tool to simulate linear viscoelastic fluid flows. The extension of LBM to the simulation of non-linear viscoelastic flows is presented here. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.