Thanks to the line-of-sight (LoS) transmission and flexibility, unmanned aerial vehicles (UAVs) effectively improve the throughput of wireless networks. Nevertheless, the LoS links are prone to severe deterioration by complex propagation environments, especially in urban areas. Reconfigurable intelligent surfaces (RISs), as a promising technique, can significantly improve channel fading effect and enhance communication quality by intelligently reflecting the received signals. Motivated by this, the joint UAV trajectory and RIS's passive beamforming design for a novel RIS-assisted UAV communication system is investigated to maximize the average achievable rate in this letter. To solve the formulated non-convex problem, we divide it into two subproblems, i.e., passive beamforming and trajectory optimization, respectively. We first derive a closed-form phaseshift solution with any given UAV trajectory to achieve the phase alignment of the received signals from different transmission paths. Then, with the given optimal phase-shift solution, we obtain the locally optimal trajectory solution by using the successive convex approximation (SCA) method. Numerical results depict that the proposed algorithm can considerably improve the average achievable rate.Index Terms-UAV communication, trajectory design, reconfigurable intelligent surface, passive beamforming.