Reliable global-position tracking control is crucial for bipedal robots to perform high-level tasks such as multi-agent coordination and dynamic-obstacle avoidance. Here, a robot’s global position refers to its position in the environment. In this study, a control approach that can provably achieve satisfactory global-position tracking is proposed for bipedal robots with a general multi-domain gait, which consists of walking phases (domains) of full actuation, over actuation, and underactuation. The derivation of the proposed control law begins with full-order modeling of the hybrid, nonlinear walking dynamics. Based on the full-order model, input-output linearizing control is synthesized, which can achieve exponential global-position tracking within the fully-actuated and the over-actuated domains but may result in uncontrolled internal dynamics within the underactuation domain. To enable reliable global-position tracking for the overall hybrid closed-loop system, the construction of multiple Lyapunov functions is employed to derive sufficient conditions under which the tracking performance of the proposed control law can be provably guaranteed. Finally, simulations of a planar bipedal robot were performed to demonstrate the effectiveness of the proposed control approach.