We consider energy and momentum transfer between surfaces in the rarefied cylindrical Couette problem. We show that in contrast to the continuum mode, in the free molecular and transient regimes the dependences of the energy and momentum fluxes transferred to the surface on the velocity of the rotating surface have maxima. This effect occurs due to the difference between the energy and momentum transfer mechanism in the continuum, transient and free molecular modes. The influence of the rarefaction on the energy and momentum transfer in the transient regime is studied using the direct simulation Monte-Carlo (DSMC) method. We show that at some surface velocities the shear stress and energy flux transferred to the surface have maxima as functions of the Knudsen number. (C) 2018 Elsevier Ltd. All rights reserved.