Flying devices, e.g., Unmanned Aerial Vehicles (UAV) and High Altitude Platforms (HAP) are showing great potentials to revolutionise human society with unprecedented efficiency and convenience. 5G and beyond (5GB) networks have been considered as an important infrastructure for supporting flying devices to accomplish mission-critical applications. However, most of the existing research on 5GB networks mainly focuses on technology evolution to support ground devices, paying insufficient attention to the emerging large-scale deployment of flying devices. To fill the gap, this study aims to identify the differences when 5GB networks are used to provide massive access services for the ground devices and their counterpart flying in the air and analyse in which aspects that 5GB should be enhanced to serve flying devices. In detail, a holistic 5GB architecture is presented to support both ground and flying devices. Then, the unique features of flying devices are analysed with a focus on the challenges they bring to 5GB systems. Facing these challenges, we thoroughly investigate the advantages and disadvantages of 5GB key technologies. Furthermore, a case study is presented to demonstrate that flying devices not only create new issues for 5GB design, but also bring new opportunities for 5GB to enhance their service capabilities.