Nusselt numbers have been calculated for heat transfer to fluids flowing through annuli under conditions of uniform heat flux and fully established velocity and temperature profiles. The following cases were considered: (a) laminar flow, (b) slug flow, (c) turbulent flow with molecular conduction only, and (d) turbulent flow with both molecular and eddy conduction. These Nusselt numbers were determined for two conditions: heat transfer from the inner wall only and heat transfer from the outer wall only. The results were correlated by semi-empirical equations. The final results obtained on cases (a), (b), and (c) are applicable to any fluid, whereas those obtained on (d) are for liquid metals only. Wall- and bulk-temperature relationships for the above four cases were also determined. These relationships were treated as dimensionless temperature ratios. Both the Nusselt numbers and temperature ratios were evaluated over the r1/r2 range, zero to unity; the former being the case of the circular pipe, and the latter, the case of infinite parallel plates.