This paper considers covert communications in the context of unmanned aerial vehicle (UAV) networks, aiming to hide a UAV for transmitting critical information out of an area that is monitored and where communication is not allowed. Specifically, the UAV as a transmitter intends to transmit information to a legitimate receiver (Bob) covertly while avoiding being detected by a warden (Willie), where location uncertainty exists at Bob and/or Willie. In order to enhance the considered covert communication performance, we jointly optimize the UAV's trajectory and transmit power in terms of maximizing the average covert transmission rate from the UAV to Bob subject to transmission outage constraint and covertness constraint. The formulated optimization problem is difficult to tackle directly due to the intractable constraints. As such, we first employ conservative approximation to transform a constraint into a deterministic form and then apply the first-order restrictive approximation to transform the optimization problem into a convex form. By applying the successive convex approximation technique, an efficient iterative algorithm is developed to solve the optimization problem. Our examination shows that the developed joint trajectory and transmit power optimization scheme achieves significantly better covert communication performance as compared to a benchmark scheme.