Dynamic time division duplex (D-TDD) dynamically allocates the transmission directions for traffic adaptation in each cell. D-TDD systems are receiving a lot of attention because they can reduce latency and increase spectrum utilization via flexible and dynamic duplex operation in 5G New Radio (NR). However, the advantages of the D-TDD system are difficult to fully utilize due to the cross-link interference (CLI) arising from the use of different transmission directions between adjacent cells. This paper is a survey of the research from academia and the standardization efforts being undertaken to solve this CLI problem and make the D-TDD system a reality. Specifically, we categorize and present the approaches to mitigating CLI according to operational principles. Furthermore, we present the signaling necessary to apply the CLI mitigation schemes. We also present information-theoretic performance analysis of D-TDD systems in various environments. As topics for future works, we discuss the research challenges and opportunities associated with the CLI mitigation schemes and signaling design in a variety of environments. This survey is recommended for those who are in the initial stage of studying D-TDD systems and those who wish to develop a more feasible D-TDD system as a baseline for reviewing the research flow and standardization trends surrounding D-TDD systems and to identify areas of focus for future works.