Unmanned aerial vehicles (UAVs) and reconfigurable intelligent surfaces (RISs) are expected to be widely used in future wireless communication networks to improve spectrum and energy efficiency. In this paper, RIS-assisted UAV communication systems are studied and analysed by developing a comprehensive mathematical framework for examining their outage performance. In order to study the effect of the RIS on UAV communications, two system scenarios are considered: in the first scenario, the UAV acts as an aerial base station (BS) serving a ground user to offload the terrestrial network, and in the second system, the UAV acts as an aerial user served by a terrestrial BS. We present channel models considering the UAV's unique characteristics, propose a closed-form approximation for the signal-to-noise-ratio (SNR) distribution, and derive an analytical expression for the relevant outage probability. Results show that RIS can significantly improve the performance of UAV communication systems by introducing energy-efficient and reliable links. This opens the door for UAV networks, which are highly scalable, adaptable, and robust to environmental changes. Furthermore, the results show that the UAV position and altitude optimisation significantly affects the outage performance.