This paper establishes a new framework for repetitive control of uncertain robot manipulators via operator-theoretic robust stabilization. After applying the inverse dynamics approach to robot manipulators, by which the relevant nonlinear input/output behavior is converted to a linear time-invariant (LTI) equation, we take the repetitive control approach. Even though such a repetitive controller is known to achieve high performances for periodic reference inputs, it is quite difficult to derive the stability analysis for the resulting closed-loop systems in a rigorous fashion. To solve this difficulty, we construct an operator-theoretic approach to the repetitive control treatment, and show that the closed-loop systems are exponentially stable if and only if the spectrum radius of the relevant monodromy operator is less than 1. Based on the necessary and sufficient condition, we develop a guideline to take the relevant control parameters. Finally, some experiment results are given to demonstrate the overall arguments developed in this paper.