Due to the broadcast nature of wireless medium, wireless communication is highly vulnerable to eavesdropping attack. Traditionally, secure wireless data transmission has relied on cryptographic techniques at the network layer which incur high computational power and complexity. As an alternative, physical layer security (PLS) is emerging as a promising paradigm to protect wireless systems by exploiting the physical characteristics of the wireless channels. Among various PLS approaches, the one based on cooperative communication is favorable and has got a lot of interest from the research community. Although PLS schemes with half-duplex relays have been extensively discovered, the issue of PLS in cooperative systems with full-duplex (FD) relays is far from being comprehensively understood. In this paper, we first present the state of the art on PLS approaches proposed for FD cooperative systems. We then provide a case study in which a source-based jamming scheme is proposed to enhance the secrecy performance of a cooperative system with an untrusted FD relay. Finally, we outline several interesting yet challenging future research problems in this topic.