Full-Duplex (FD) wireless technology enables a radio to transmit and receive on the same frequency band at the same time, and it is considered to be one of the candidate technologies for the fifth generation (5G) of wireless communications due to its advantages including potential doubling of the capacity, reduced end-to-end and feedback delays, improved network secrecy and efficiency, and starting with a detailed overview of FD-enabled DSS, we provide a comprehensive survey of recent advances in this domain. We then highlight several potential techniques for enabling FD operation in DSS wireless systems. Subsequently, we propose a novel communication framework to enable CST in DSS systems by employing a power control-based SI mitigation scheme and carry out the throughput performance analysis of this proposed framework. Finally, we discuss some open research issues and future directions with the objective of stimulating future research efforts in the emerging FD-enabled DSS wireless systems.