We present a comprehensive review for wireless power transfer (WPT)-aided full-duplex (FD) relay systems. Two critical challenges in implementing WPT-aided FD relay systems are presented, that is, pseudo FD realization and high power consumption. Existing time-splitting or power-splitting structure based-WPT-aided FD relay systems can only realize FD operation in one of the time slots or only forward part of the received signal to the destination, belonging to pseudo FD realization. Besides, self-interference is treated as noise and self-interference cancellation (SIC) operation incurs high power consumption at the FD relay node. To this end, a promising solution is outlined to address the two challenges, which realizes consecutive FD realization at all times and forwards all the desired signal to the destination for decoding. Also, active SIC, that is, analog/digital cancellation, is not required by the proposed solution, which effectively reduces the circuit complexity and releases high power consumption at the FD relay node. Specific classifications and performance metrics of WPT-aided FD relay systems are summarized. Some future research is also envisaged for WPT-aided FD systems.