This article investigates the physical-layer security of cognitive radio (CR) networks, which are vulnerable to various newly arising attacks targeting on the weaknesses of CR communications and networking. We first review a range of physical-layer attacks in CR networks, including the primary user emulation, sensing falsification, intelligence compromise, jamming and eavesdropping attacks. Then we focus on the physical-layer security of CR networks against eavesdropping and examine the secrecy performance of cognitive communications in terms of secrecy outage probability. We further consider the use of relays for improving the CR security against eavesdropping and propose an opportunistic relaying scheme, where a relay node that makes CR communications most resistant to eavesdropping is chosen to participate in assisting the transmission from a cognitive source to its destination. It is illustrated that the physical-layer secrecy of CR communications relying on the opportunistic relaying can be significantly improved by increasing the number of relays, showing the security benefit of exploiting relay nodes. Finally, we present some open challenges in the field of relays assisted physical-layer security for CR networks.