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In recent years, unmanned aerial vehicles (UAVs) have received considerable attention from regulators, industry and research community, due to rapid growth in a broad range of applications. Particularly, UAVs are being used to provide a promising solution to reliable and cost-effective wireless communications from the sky. The deployment of UAVs has been regarded as an alternative complement of existing cellular systems, to achieve higher transmission efficiency with enhanced coverage and capacity. However, heavily utilized microwave spectrum bands below 6 GHz utilized by legacy wireless systems are insufficient to attain remarkable data rate enhancement for numerous emerging applications. To resolve the spectrum crunch crisis and satisfy the requirements of 5G and beyond mobile communications, one potential solution is to use the abundance of unoccupied bandwidth available at millimeter wave (mmWave) frequencies. Inspired by the technique potentials, mmWave communications have also paved the way into the widespread use of UAVs to assist wireless networks for future 5G and beyond wireless applications. In this paper, we provide a comprehensive survey on current achievements in the integration of 5G mmWave communications into UAV-assisted wireless networks. More precisely, a taxonomy to classify the existing research issues is presented, by considering seven cutting-edge solutions. Subsequently, we provide a brief overview of 5G mmWave communications for UAV-assisted wireless networks from two aspects, i.e., key technical advantages and challenges as well as potential applications. Based on the proposed taxonomy, we further discuss in detail the state-of-the-art issues, solutions, and open challenges for this newly emerging area. Lastly, we complete this survey by pointing out open issues and shedding new light on future directions for further research on this area.