In this paper, an improved modified three-step iteration method is introduced and analysed for fixed points, as well as common fixed points, of asymptotically nonepansive operators in uniformly smooth and convex Banach spaces. Our results generalize many relevant results in the literature, including those of Schu [11], Osilike and Aniagbosor [6], Xu and Noor [14], and Owojori and Imoru [8].