Unmanned aerial vehicles (UAVs) have been widely used in both military and civilian applications, where a stable communication link is vital for safe flight control and robust data transmission. To develop a reliable UAV communication system, it is necessary to deeply understand the UAV channel characteristics and establish accurate channel models. Channel sounding is the most effective way to obtain realistic channel characteristics and validate the theoretical channel model. However, the studies on UAV channel sounding are still insufficient in terms of system design and data processing due to the complexity of developing a UAV channel sounder. Different from the terrestrial channel sounders, the implementation of a UAV channel sounder is tortured by the limited battery life and payload capacity of the UAV platform. The sounding scheme and data post-processing also need to be specially designed for highly dynamic UAV channels. So far, most existing survey studies on UAV channels focus on modeling methodology and model presentation. To fill this gap, this paper provides a comprehensive survey on the design of the UAV channel sounder, in terms of the hardware scheme, sounding signal, time synchronization, calibration, and data postprocessing. Current issues and potential research topics behind existing sounding technologies and measurement campaigns are analyzed. Moreover, future challenges and open issues are also discussed.