Abstract In this paper we use numerical simulations to study two-dimensional steady flows of a viscoelastic fluid past a circular cylinder confined by two parallel walls. The drag on the cylinder and the velocity profile in its wake are investigated as functions of the wall blockage and properties of the fluid. The interplay among wall effects, elasticity, shear-thinning and inertia is examined in detail. Results show that wall proximity shortens the wake and increases the drag, and this effect is reduced by fluid elasticity. For weak wall blockage, elasticity increases the drag and lengthens the wake for Reynolds number Re=0.1 to 10. For stronger blockage this trend is reversed. Shear-thinning decreases the drag and shortens the wake for all Reynolds numbers, Weissenberg numbers and blockage ratios we have tested. A negative wake appears for the strongest wall blockage.