Summary: Turbulent Taylor-Couette flow with arbitrary rotation frequencies \(\omega_1, \omega_2\) of the two coaxial cylinders with radii \(r_1 < r_2\) is analysed theoretically. The current \(J^{\omega}\) of the angular velocity \(\omega (\mathbf {x},t) = u_{\phi}(r,\phi ,z,t)/r\) across the cylinder gap and and the excess energy dissipation rate \(\in_w\) due to the turbulent, convective fluctuations (the ‘wind’) are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor-Couette flow with thermal Rayleigh-Bénard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio \(\eta = r_1/r_2\) or the gap width \(d = r_2 - r_1\) between the cylinders. A quantity \(\sigma\) corresponding to the Prandtl number in Rayleigh-Bénard flow can be introduced, \(\sigma = ((1 + \eta)/2)/\sqrt{\eta})^4\). In Taylor-Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh-Bénard flow. The analogue of the Rayleigh number is the Taylor number, defined as \(Ta \propto (\omega_1 - \omega_2)^2\) times a specific geometrical factor. The experimental data show no pure power law, but the exponent \(\alpha\) of the torque versus the rotation frequency \(\omega_1\) depends on the driving frequency \(\omega_1\). An explanation for the physical origin of the \(\omega_1\)-dependence of the measured local power-law exponents \(\alpha (\omega_1)\) is put forward. Also, the dependence of the torque on the gap width \(\eta\) is discussed and, in particular its strong increase for \(\eta \to 1\).