Millimeter wave (mmWave) communications have recently attracted large research interest, since the huge available bandwidth can potentially lead to rates of multiple Gbps (gigabit per second) per user. Though mmWave can be readily used in stationary scenarios such as indoor hotspots or backhaul, it is challenging to use mmWave in mobile networks, where the transmitting/receiving nodes may be moving, channels may have a complicated structure, and the coordination among multiple nodes is difficult. To fully exploit the high potential rates of mmWave in mobile networks, lots of technical problems must be addressed. This paper presents a comprehensive survey of mmWave communications for future mobile networks (5G and beyond). We first summarize the recent channel measurement campaigns and modeling results. Then, we discuss in detail recent progresses in multiple input multiple output (MIMO) transceiver design for mmWave communications. After that, we provide an overview of the solution for multiple access and backhauling, followed by analysis of coverage and connectivity. Finally, the progresses in the standardization and deployment of mmWave for mobile networks are discussed.