A robot localization system is presented that enables a robot to estimate its position within some space by passively receiving ultra-wideband radio signals from fixed-position modules. Communication from the fixed-position modules is one-way, allowing the system to scale to multiple robots. Furthermore, the system's high position update rate makes it suitable to be used in a feedback control system, and enables the robot to track and perform high-speed, dynamic motions. This paper describes the algorithmic underpinnings of the system, discusses design decisions and their impact on the performance of the resulting localization, and highlights challenges faced during implementation. Performance of the localization system is experimentally verified through comparison with data from a motion-capture system. Finally, the system's application to robot self-localization is demonstrated through integration with a quadrocopter.