In the present work, we consider the linear hydrodynamic stability problems of viscoelastic fluids in arbitrary finite domains. The effects of domain shapes on the critical Rayleigh number and convection pattern are investigated by means of a linear stability analysis employing a Chebyshev pseudospectral method. It is shown that the domain shape can change the viscoelastic parameter values where the Hopf bifurcation occurs in the Rayleigh-Benard convection. The results of the present investigation may be exploited to design shapes of convection box where the Hopf bifurcation occurs at realistic low values of Deborah number. This will enhance the usefulness of the natural convection system as a rheometry tool. (C) 2003 Elsevier Ltd. All rights reserved.