Measurements of the Nusselt number Nu and of properties of the large-scale circulation (LSC) for turbulent Rayleigh–Bénard convection are presented in the presence of rotation about a vertical axis at angular speeds 0 ≤ Ω ≲ 2 rad s−1. The sample chamber was cylindrical with a height equal to the diameter, and the fluid contained in it was water. The LSC was studied by measuring sidewall temperatures as a function of azimuthal position. The measurements covered the Rayleigh-number range 3 × 108 ≲ Ra ≲ 2 × 1010, the Prandtl-number range 3.0 ≲ Pr ≲ 6.4 and the Rossby-number range 0 ≤ (1/Ro ∝ Ω) ≲ 20. At modest 1/Ro, we found an enhancement of Nu due to Ekman-vortex pumping by as much as 20%. As 1/Ro increased from zero, this enhancement set in discontinuously at and grew above 1/Roc. The value of 1/Roc varied from about 0.48 at Pr = 3 to about 0.35 at Pr = 6.2. At sufficiently large 1/Ro (large rotation rates), Nu decreased again, due to the Taylor–Proudman (TP) effect, and reached values well below its value without rotation. The maximum enhancement increased with increasing Pr and decreasing Ra and, we believe, was determined by a competition between the Ekman enhancement and the TP depression. The temperature signature along the sidewall of the LSC was detectable by our method up to 1/Ro ≃ 1. The frequency of cessations α of the LSC grew dramatically with increasing 1/Ro, from about 10−5 s−1 at 1/Ro = 0 to about 2 × 10−4 s−1 at 1/Ro = 0.25. A discontinuous further increase of α, by about a factor of 2.5, occurred at 1/Roc. With increasing 1/Ro, the time-averaged and azimuthally averaged vertical thermal gradient along the sidewall first decreased and then increased again, with a minimum somewhat below 1/Roc. The Reynolds number of the LSC, determined from oscillations of the time correlation functions of the sidewall temperatures, was constant within our resolution for 1/Ro ≲ 0.3 and then decreased with increasing 1/Ro. The retrograde rotation rate of the LSC circulation plane exhibited complex behaviour as a function of 1/Ro even at small rotation rates corresponding to 1/Ro < 1/Roc.