Millimeter-wave (mmWave) communication technology offers a potential and promising solution to support 5G and B5G wireless networks in dynamic scenarios and applications. However, mobility introduces many challenges as well as opportunities to mmWave applications. To address these problems, we conduct a survey of the opportunities and technologies to support mmWave communications in mobile scenarios. Firstly, we summarize the mobile scenarios where mmWave communications are exploited, including

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... indoor wireless local area network (WLAN) or wireless personal area network (WPAN), cellular access, vehicle-to-everything (V2X), high speed train (HST), unmanned aerial vehicle (UAV), and the new space-air-ground-sea communication scenarios. Then, to address users' mobility impact on the system performance in different application scenarios, we introduce several representative mobility models in mmWave systems, including human mobility, vehicular mobility, high speed train mobility and ship mobility. Next we survey the key challenges and existing solutions to mmWave applications, such as channel modeling, channel estimation, anti-blockage, and capacity improvement. Lastly, we discuss the open issues concerning mobility-aware mmWave communications that deserve further investigation. In particular, we highlight future heterogeneous mobile networks, dynamic resource management, artificial intelligence (AI) for mobility and integration of geographical information, deployment of large intelligent surface and reconfigurable antenna technology, and finally, the evolution to Terahertz (THz) communications.