Person re-identification is a key challenge for surveillance across multiple sensors. Prompted by the advent of powerful deep learning models for visual recognition, and inexpensive RGBD cameras and sensor-rich mobile robotic platforms, e.g. self-driving vehicles, we investigate the relatively unexplored problem of cross-modal re-identification of persons between RGB (color) and depth images. The considerable divergence in data distributions across different sensor modalities introduces additional challenges to the typical difficulties like distinct viewpoints, occlusions, and pose and illumination variation. While some work has investigated re-identification across RGB and infrared, we take inspiration from successes in transfer learning from RGB to depth in object detection tasks. Our main contribution is a novel cross-modal distillation network for robust person re-identification, which learns a shared feature representation space of person's appearance in both RGB and depth images. The proposed network was compared to conventional and deep learning approaches proposed for other cross-domain re-identification tasks. Results obtained on the public BIWI and RobotPKU datasets indicate that the proposed method can significantly outperform the state-of-the-art approaches by up to 10.5% mAp, demonstrating the benefit of the proposed distillation paradigm.