This paper presents a catadioptric omnidirectional thermal odometry (COTO) system that estimates the six degrees of freedom (DoF) pose of a camera using only omnidirectional thermal images in visually degraded, fast-motion, and dynamic environments. First, we design and fabricate a central hyperbolic catadioptric omnidirectional thermal camera that captures surrounding thermal images with 360° horizontal field of view (FoV), and improve the omnidirectional camera model and calibration method to obtain high-precision camera intrinsic parameter. Second, we propose the epipolar curve constraint combining with omnidirectional thermal object detection to significantly reduce the interference of moving objects on pose estimation. Third, the implemented COTO pipeline consists of photometric calibration, dynamic region removal, tracking and mapping to overcome the drawbacks of photometric inconsistency and large distortion in omnidirectional thermal images. Experiments have been conducted on a total of 17 sequences of Lab, Outdoor and Driving, amounting to more than 60,000 omnidirectional thermal images of real environments. The experimental results indicate that the proposed COTO system has excellent localization accuracy and unparalleled robustness over the current state-of-the-art methods. The average localization accuracy measured by the absolute trajectory error (ATE) is less than 15 cm from the ground truth in both Lab and Outdoor sequences. In addition, COTO was the only system with complete and successful tracking in all sequences. The system can be used as an innovative localization solution, particularly in challenging environments with changes in ambient light, rapid vehicle motion, and moving object interference, which can be a difficult problem for visual odometry to solve.