A numerical study on the transient flow of a viscoelastic fluid is presented. The numerical framework is that of Smoothed Particle Hydrodynamics (SPH) already used by Ellero et al. in previous simulations of Non-Newtonian flows [J. Non-Newtonian Fluid. Mech. 105 (2002) 35-51]. In particular, the start-up flow between parallel plates is simulated for an Oldroyd-B and UCM fluid at low Reynolds number. Results for a Newtonian fluid are also shown for comparison. The numerical results are presented and compared with available theoretical solutions, showing a very good agreement. In particular, the simulations of an Oldroyd-B fluid have been found to be stable and accurate for a wide range of the Weissenberg number. In the case of a UCM fluid, the absence of a viscous term in the momentum equation makes its numerical modelling harder. Namely, the process is characterised by a travelling damped wave, which, if not accurately resolved. can lead to the rapid growth of small oscillations in time eventually Causing divergence. On the other hand. if specifically dealing with transient flow problems, stabilising techniques such as BSD, EVSS or AVSS can not be used either; whenever used in conjunction with decoupled solution algorithms, they give an excessive oversmoothing in the results which deteriorates the final accuracy. In this work, we consider an exact SPH discretisation of the hyperbolic equation characterising the UCM model. SPH simulations are finally performed for different Weissenberg numbers showing very promising results. Finally, a discussion on the SPH treatment of the boundary conditions for general hydrodynamics problems is also outlined following the approach of ' SPH boundary particles ' introduced by Morris for the simulations of low Reynolds number flows. (c) 2005 Elsevier B.V. All rights reserved.