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Fifth-generation (5G) and beyond mobile communication networks are expected to meet an explosion of data traffic usage and a fast-varying environment. The millimetre-wave communications and unmanned aerial vehicles (UAVs) communications are two important methods to tackle these challenges. To thoroughly investigate millimetre-wave UAV communications, it is essential to have a good understanding of electromagnetic wave propagation in the millimetre-wave band between the UAV-carried aerial base station or the mobile relay node and ground nodes, which is known as the UAV air-to-ground (A2G) channel model. To support the millimetre-wave UAV A2G network design, it is vital to have a deep cognition of the network performance evaluation parameters of the UAV A2G link, e.g., throughput and energy efficiency. This thesis discusses three problems related to millimetre-wave UAV A2G communications. In this study, the effect of the inevitable UAV wobbling on the millimetre-wave UAV A2G channel is first investigated. The wobbling process of a hovering UAV, which is affected by wind gusts and the high vibration frequency of its propellers and rotors, is modelled. The analytical temporal autocorrelation function (ACF) for the millimetre-wave UAV A2G link is derived. With the derived temporal ACF equation, the Doppler power spectrum density for the millimetre-wave UAV A2G link is investigated. The numerical results show that the temporal ACF decreases quickly with time and the impact of the Doppler effect caused by UAV wobbling is significant on bit error probability (BEP) for the millimetre-wave A2G link. Then, the problem of throughput for the millimetre-wave UAV A2G link under UAV wobbling is investigated. Two types of detectors at the receiver to demodulate the received signal and get the instantaneous BEP of a millimetre-wave UAV A2G link under UAV wobbling are introduced. Based on the designed detectors, an adaptive modulation scheme maximising the average transmission rate under UAV wobbling by optimizing the data transmission time subject to the maximum tolerable BEP is proposed. The numerical results show that the proposed adaptive modulation maximises the temporally averaged transmission rate of the millimetre-wave UAV A2G link compared with other transmission policies under UAV wobbling. After proposing the adaptive modulation, the power control to minimise the power consumption is investigated considering the limited on-board energy of a UAV. A power control policy that minimises the transmission power while maintaining both the BEP under the threshold and the maximised average transmission rate is proposed for the millimetre-wave UAV A2G link under UAV wobbling. The energy efficiency of the UAV A2G link is evaluated to show how effective this power control policy is. The numerical results show that the power control policy reduces the power consumption by up to 50% for wobbling millimetre-wave UAV A2G links and the energy efficiency of the system under power control is higher than that of the adaptive modulation scheme without the power control policy. In summary, the thesis studies the channel characteristics and evaluates the performance of the millimetre-wave UAV A2G link under wobbling to support the future millimetre-wave UAV communication network deployment. A key observation is that even for weak UAV wobbling, the temporal ACF of the UAV A2G link deteriorates quickly, making the link difficult to establish a reliable communication link. To keep the reliable A2G link and achieve high throughput, the adaptive modulation scheme of the millimetre-wave UAV A2G link under wobbling is proposed. The power control policy for the adaptive modulation of the millimetre-wave UAV A2G link could save power by over 50% and support the green UAV A2G link.