In this paper, we investigate robust resource allocation algorithm design for multiuser downlink multiple-input single-output (MISO) unmanned aerial vehicle (UAV) communication systems, where we account for the various uncertainties that are unavoidable in such systems and, if left unattended, may severely degrade system performance.We jointly optimize the two-dimensional (2-D) trajectory and the transmit beamforming vector of the UAV for minimization of the total power consumption. The algorithm design is formulated as a non-convex optimization problem taking into account the imperfect knowledge of the angle of departure (AoD) caused by UAV jittering, user location uncertainty, wind speed uncertainty, and polygonal no-fly zones (NFZs). Despite the non-convexity of the optimization problem, we solve it optimally by employing monotonic optimization theory and semidefinite programming relaxation which yields the optimal 2-D trajectory and beamforming policy. Since the developed optimal resource allocation algorithm entails a high computational complexity, we also propose a suboptimal iterative low-complexity scheme based on successive convex approximation to strike a balance between optimality and computational complexity. Our simulation results reveal not only the significant power savings enabled by the proposed algorithms compared to two baseline schemes, but also confirm their robustness with respect to UAV jittering, wind speed uncertainty, and user location uncertainty. Moreover, our results unveil that the joint presence of wind speed uncertainty and NFZs has a considerable impact on the UAV trajectory. Nevertheless, by counteracting the wind speed uncertainty with the proposed robust design, we can simultaneously minimize the total UAV power consumption and ensure a secure trajectory that does not trespass any NFZ. ). This paper was presented in part at IEEE Globecom 2018 [1]. 2 be employed as aerial base stations to offer temporary communication links in a timely and cost-effective manner. Moreover, due to their high mobility and maneuverability, UAVs can adapt their trajectories based on the actual environment and terrain which improves system performance [3]. As a result, UAV-assisted communication systems have drawn significant attention from both academia and industry. For instance, the authors of [4] studied suboptimal UAV trajectory design for maximization of the energy-efficiency of UAV communication systems. The authors of [5] proposed a suboptimal joint trajectory, power allocation, and user scheduling algorithm for maximization of the minimum user throughput in multi-UAV systems.Secure UAV communications was investigated in [6] where the trajectory of a UAV and its transmit power were jointly optimized to maximize the system secrecy rate. The authors of [7] proposed solar-powered UAV communication systems and studied the jointly optimal resource allocation and UAV trajectory design for maximization of the system sum throughput. In fact, the throughput of UAV communication systems can be further i...