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Full-duplex (FD) transmission has already been regarded and developed as a promising method to improve the utilization efficiency of the limited spectrum resource, as transmitting and receiving are allowed to simultaneously occur on the same frequency band. Nowadays, benefiting from the recent development of intelligent reflecting surface (IRS), some unique electromagnetic (EM) functionalities, like wavefront shaping, focusing, anomalous reflection, absorption, frequency shifting, and nonreciprocity can be realized by soft-controlled elements at the IRS, showing the capability of reconfiguring the wireless propagation environment with no hardware cost and nearly zero energy consumption. To jointly exploit the virtues of both FD transmission and IRS, in this article we first introduce several EM functionalities of IRS that are profitable for FD transmission; then, some designs of FD-enabled IRS systems are proposed and discussed, followed by numerical results to demonstrate the obtained benefits. Finally, the challenges and open problems of realizing FD-enabled IRS systems are outlined and elaborated upon.