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.