Existing experimental data on planar and axisymmetric contraction flows of constant-viscosity and shear-thinning elastic liquids have presented rheologists with some provocative challenges. some of which we hope to discuss in this paper by carrying out numerical simulations for various constitutive models. Amongst the questions we hope to address are the following: 1. Why do existing numerical simulations for the Oldroyd B model fail to predict the significant increases observed in the Couette correction for Boger fluids? 2. What are the roles of 'normal stress differences' and 'extensional viscosity' in determining the Couette correction? 3. What is the future for contraction-flow experiments in providing a measure of 'resistance to stretch'? In order to make headway, we shall concentrate on the axisymmetric contraction-expansion 4:1:4 geometry with rounded corners, which from a numerical standpoint is easier to handle than the more popular 4:1 contraction geometry with sharp corners. (There is sufficient evidence in the experimental literature to indicate that the contraction-expansion geometry shows many of the features found in the 4:1 contraction geometry.) (C) 2009 Elsevier Ltd. All rights reserved.