Axial flow and heat transfer in a horizontal cylindrical annulus can be influenced by eccentricity of the inner cylinder and the presence of buoyancy forces. A numerical study is presented for the combined forced and free convection for the fully developed flow and heat transfer for eccentric annuli of different eccentricities and radius ratios. The flow field is characterized by large cross-stream secondary currents and significant flow distortion. The Nusselt number increases significantly with the Rayleigh number; the corresponding increase in the friction factor is relatively small. The eccentricity introduces additional nonuniformity of the flow and temperature fields.