Although the application of the fifth-Generation (5G) mobile communication has brought tremendous innovations to the daily life of human beings, e.g., autonomous vehicles and internet of everything, the upcoming huger data requirement leads to the emergence of the sixth-Generation (6G) mobile communication. Compared to 5G, the transmission rate, time delay, and wireless coverage need to be improved significantly. Thus, in this paper the applications of Unmanned Aerial Vehicles (UAVs) to the ubiquitous, intelligent and coupling 6G network are surveyed. First, the utilization of UAVs in the framework of space-air-ground-sea integrated network is demonstrated, and the roles and functions of UAVs in different scenarios are emphasized, e.g., the swarm base stations, the deployment for holographic projection, the long-distance relaying and the data collection. Then, the potential 6G key techniques of terahertz, ultra-massive multiple-input and multiple-output, endogenous artificial intelligence, Intelligent Reflecting Surface (IRS), intelligent edge computing, blockchain and integrated sensing and communication for UAV communications are investigated. Finally, the future challenges of UAV communications for 6G, including the limited duration, integration of networks, compatibility of IRS, development of THz communications, and user security are discussed.