Millimeter wave (mmWave) communication has raised increasing attentions from both academia and industry due to its exceptional advantages. Compared with existing wireless communication techniques, such as WiFi and 4G, mmWave communications adopt much higher carrier frequencies and thus come with advantages including huge bandwidth, narrow beam, high transmission quality, and strong detection ability. These advantages can well address difficult situations caused by recent popular applications using wireless technologies. For example, mmWave communications can significantly alleviate the skyrocketing traffic demand of wireless communication from video streaming. Meanwhile, mmWave communications have several natural disadvantages, e.g., severe signal attenuation, easily blocked by obstacles, and small coverage, due to its short wavelengths. Hence, the major challenge is how to overcome its shortcomings while fully utilizing its advantages. In this paper, we present a taxonomy based on the layered model and give an extensive review on mmWave communications. Specially, we divide existing efforts into four categories that investigate: physical layer, medium access control (MAC) layer, network layer, and cross layer optimization, respectively. First, we present an overview of some technical details in physical layer. Second, we summarize available literature in MAC layer that pertains to protocols and scheduling schemes. Third, we make an in-depth survey of related research work in network layer, providing brain storming and methodology for enhancing the capacity and coverage of mmWave networks. Fourth, we analyze available research work related to cross layer allocation/optimization for mmWave communications. Fifth, we make a review of mmWave applications to illustrate how mmWave technology can be employed to satisfy other services. At the end of each section described above, we point out the inadequacy of existing work and identify the future work. Sixth, we present some available resources for mmWave communications, including related books about mmWave, commonly used mmWave frequencies, existing protocols based on mmWave, and experimental platforms. Finally, we have a simple summary and point out several promising future research directions.