Due to its high mobility and flexible deployment, unmanned aerial vehicle (UAV) is drawing unprecedented interest in both military and civil applications to enable agile wireless communications and provide ubiquitous connectivity. Mainly operating in an open environment, UAV communications can benefit from dominant line-of-sight links; however, it on the other hand renders the UAVs more vulnerable to malicious eavesdropping or jamming attacks. Recently, physical layer security (PLS), which exploits the inherent randomness of the wireless channels for secure communications, has been introduced to UAV systems as an important complement to the conventional cryptography-based approaches. In this paper, a comprehensive survey on the current achievements of the UAV-aided wireless communications is conducted from the PLS perspective. We first introduce the basic concepts of UAV communications including the typical static/mobile deployment scenarios, the unique characteristics of air-to-ground channels, as well as various roles that a UAV may act when PLS is concerned. Then, we introduce the widely used secrecy performance metrics and start by reviewing the secrecy performance analysis and enhancing techniques for statically deployed UAV systems, and extend the discussion to a more general scenario where the UAVs' mobility is further exploited. For both cases, respectively, we summarize the commonly adopted methodologies in the corresponding analysis and design, then describe important works in the literature in detail. Finally, potential research directions and challenges are discussed to provide an outlook for future works in the area of UAV-PLS in 5G and beyond networks.