In this study, the combined influence of fluid viscoelasticity and inertia on the flow and heat transfer characteristics of a circular cylinder in the steady laminar flow regime have been studied numerically. The momentum and energy equations together with an appropriate viscoelastic constitutive equation have been solved numerically using the finite volume method over the following ranges of conditions: Reynolds number, ; elasticity number (= Wi/Re, where Wi is the Weissenberg number), ; Prandtl number, for Oldroyd-B and finitely extensible nonlinear elastic-Peterlin (FENE-P) (with two values of the chain extensibility parameter L2, namely 10 and 100) viscoelastic fluid models including the limiting case of Newtonian fluids (El = 0). New extensive results are presented and discussed in terms of the streamline and isotherm profiles, drag coefficient, distribution of the local and surface averaged Nusselt number. Within the range of conditions embraced here, the separation of boundary layers (momentum and thermal) is seen to be completely suppressed in an Oldroyd-B fluid whereas it is accelerated for a FENE-P fluid in comparison with that seen for a Newtonian fluid otherwise under identical conditions. At a fixed elasticity number, both the drag coefficient and average Nusselt number are seen to be independent of the Reynolds number beyond a critical value for an Oldroyd-B fluid. In contrast, the drag coefficient decreases and the average Nusselt number increases with Reynolds number for a FENE-P fluid at a constant value of the elasticity number. Finally, a simple correlation for the average Nusselt number for a FENE-P fluid is presented which facilitates the interpolation of the present results for the intermediate values of the governing parameters and/or its a priori estimation in a new application.