The paper provides a comprehensive survey of millimeter wave (mmWave) communications for future mobile networks, focusing on 5G and beyond. It begins by summarizing recent channel measurement campaigns and modeling results, followed by detailed discussions on multiple input multiple output (MIMO) transceiver design for mmWave communications. The paper then covers multiple access and backhauling solutions, coverage and connectivity analysis, and the progress in standardization and deployment of mmWave for mobile networks. Key challenges and technical potentials of mmWave communications in mobile networks are highlighted, including severe pathloss, high penetration loss, high power consumption, and hardware impairments. The paper also discusses the technical advantages of mmWave, such as large continuous unused bandwidth and short wavelengths facilitating compact antenna arrays. Channel modeling and MIMO design for mmWave communications are detailed, including the choice of MIMO architectures and channel estimation techniques. The paper concludes with a discussion on the future research directions and challenges in mmWave communications for mobile networks.The paper provides a comprehensive survey of millimeter wave (mmWave) communications for future mobile networks, focusing on 5G and beyond. It begins by summarizing recent channel measurement campaigns and modeling results, followed by detailed discussions on multiple input multiple output (MIMO) transceiver design for mmWave communications. The paper then covers multiple access and backhauling solutions, coverage and connectivity analysis, and the progress in standardization and deployment of mmWave for mobile networks. Key challenges and technical potentials of mmWave communications in mobile networks are highlighted, including severe pathloss, high penetration loss, high power consumption, and hardware impairments. The paper also discusses the technical advantages of mmWave, such as large continuous unused bandwidth and short wavelengths facilitating compact antenna arrays. Channel modeling and MIMO design for mmWave communications are detailed, including the choice of MIMO architectures and channel estimation techniques. The paper concludes with a discussion on the future research directions and challenges in mmWave communications for mobile networks.