We report high-precision measurements of the Nusselt number as a function of the Rayleigh number in water-filled rectangular Rayleigh–Bénard convection cells. The horizontal length and width of the cells are 50.0 and 15.0 cm, respectively, and the heights , 25.0, 12.5, 6.9, 3.5, and 2.4 cm, corresponding to the aspect ratios , , , , , and . The measurements were carried out over the Rayleigh number range and the Prandtl number range . Our results show that for rectangular geometry turbulent heat transport is independent of the cells’ aspect ratios and hence is insensitive to the nature and structures of the large-scale mean flows of the system. This is slightly different from the observations in cylindrical cells where is found to be in general a decreasing function of , at least for and larger. Such a difference is probably a manifestation of the finite plate conductivity effect. Corrections for the influence of the finite conductivity of the top and bottom plates are made to obtain the estimates of for plates with perfect conductivity. The local scaling exponents of are calculated and found to increase from 0.243 at to 0.327 at .