The fifth-generation (5G) wireless communication has experienced an upsurge of interest for empowering vertical industries, due to its high data volume, extremely low latency, high reliability, and significant improvement in user experience. Specifically, deploying 5G on high-speed railway (HSR) is critical for the promotion of smart travelling such that passengers can connect to the Internet and utilize the on-board time to continue their usual activities. However, there remains a series of challenges in practical implementation, such as the serious Doppler shift caused by the high mobility, the carriage penetration loss especially in the high-frequency bands, frequent handovers, and economic issues. To address these challenges, we propose three schemes in this article to improve the coverage of 5G networks on the train. In particular, we provide a comprehensive description of each scheme in terms of their network architecture and service establishment procedures. Specifically, the mobile edge computing (MEC) is used as the key technology to provide low-latency services for on-board passengers. Moreover, these three schemes are compared among themselves regarding the quality-of-service, the scalability of service, and the related industry development status. Finally, we discuss various potential research directions and open issues in terms of deploying 5G networks on HSR.