Integrating unmanned aerial vehicles (UAVs) into future wireless systems such as the fifth-generation (5G) cellular network is anticipated to bring significant benefits for both UAV and telecommunication industries. Generally speaking, UAVs can be used as new aerial platforms in the cellular network to provide communication services for terrestrial users, or become new aerial users of the cellular network served by the terrestrial base stations. Due to their high altitude, UAVs usually have dominant line-of-sight (LoS) channels with the ground nodes, which, however, pose new security challenges to future wireless networks with widely deployed UAVs. On one hand, UAV-ground communications are more prone than terrestrial communications to eavesdropping and jamming attacks by malicious nodes on the ground. On the other hand, compared to malicious ground nodes, malicious UAVs can launch more effective eavesdropping and jamming attacks to terrestrial communications. Motivated by the above, in this article, we aim to identify such new issues from a physical-layer security viewpoint and propose novel solutions to tackle them efficiently. Numerical results are provided to validate their effectiveness and promising directions for future research are also discussed.