Steady and unsteady flows between rotating cylinders are of interest on lubrication, convective heat transfer and flow-induced vibration in large rotating machinery. Steady rotating flow is generated by rotating cylinder with constant velocity while the unsteady rotating flow by oscillating cylinder with homogeneoysly oscillating velocity. An analytical method is developed based on the simple radial coordinate transformation for the steady and unsteady rotating flows in concentric annulus. The governing equations are simplified from Navier-Stokes equatins. Considering the skin friction based on the radial variation of circumferential flow velocity, the torques acting on the fixed and the rotating cylinder are evaluated in terms of added-inertia and added-damping torque coefficients. The coefficients are found to be influenced by the oscillatory Reynolds number and the radius ratio of two cylinders; however, the effect of the oscillatory Reynolds number on the coefficients is minor in case of relatively low radius ratio.