A backstepping-based predefined-time repetitive learning control scheme is proposed for uncertain robot manipulators to achieve rapid and high-precision tracking control performance. A non-singular predefined-time virtual controller is constructed to effectively avoid the singularity issues caused by the differentiation of the virtual controller in conventional finite-time backstepping design. It ensures that the tracking error of the robot manipulators joint positions converges to a neighborhood of the origin within the predefined time. Then, the lumped uncertainty of the manipulator is separated into periodic and non-periodic parts by considering the periodic characteristics of the desired trajectory. A fully saturated repetitive learning law is constructed to accurately estimate and compensate for the periodic uncertainty. Meanwhile, a robust control law is developed and the terminal attracting technique is applied to guarantee the effective compensation of the non-periodic uncertainty including external disturbances, such that the high-precision tracking of the robot manipulators joint positions is achieved. Finally, the stability of the closed-loop system and the error convergence performance of the proposed scheme are analyzed through the Lyapunov stability synthesis. The effectiveness of the proposed control method is verified by comparative simulations.