This paper presents the effects of eccentricity on fluid flow in small (less than 0.3), medium (0.3-0.6) and large (0.6-0.8) radii ratios. The Couette-Poiseuille flow, in concentric and eccentric annuli with a rotating inner pipe, is investigated. Effects of eccentricity (0.0, 0.15, 0.30, 0.45, 0.6, 0.75 and 0.90) on the flow structure, velocity distribution and pressure drop are investigated at radii ratios of 0.2, 0.4, 0.6 and 0.8 for Reynolds number from 9780 to 288623 and Taylor number from 0 to 4.39×107. The results demonstrate that the pressure drop decreases significantly with increasing eccentricity, at radius ratio 0.8. But, the effect of eccentricity on the pressure drop is negligible at radius ratios lower than 0.6. As wide annuli are less sensitive to the rotation of the inner cylinder than the narrow annuli, Taylor vortices were not established under the investigated parameters as a result of dominated axial velocity over tangential velocity in the considered range of parameters. The effect of eccentricity on the velocity becomes more influential as the radii ratio increases (small annuli) and vice versa. Furthermore, the effect of eccentricity is more influential on the axial and tangential velocity distributions, in the small space of the eccentric annulus. Axial velocity decreases while tangential velocity increases as the eccentricity increased.