This review focuses on the aerial transportation of suspended-cable payloads using quadrotors. It provides an overview of the current state of the art in this field, covering the last sixteen years of research. The paper discusses various dynamic cable models and control systems used for both individual and collaborative transport scenarios. It highlights the challenges associated with modeling and controlling the nonlinear dynamics of quadrotor systems with suspended payloads, as well as the different strategies employed to address these challenges. The review also emphasizes the importance of cable modeling in achieving accurate and stable flight performance, and the need for robust control strategies to handle the complexities introduced by the suspended load. The paper discusses the advantages and limitations of different cable models, including taut-cable, elastic-cable, and flexible-cable models, and their respective applications in both individual and collaborative transport scenarios. It also explores the various control strategies used in the field, including anti-swing control, optimal trajectory following, and control for aggressive maneuvers. The review concludes with a discussion of the main trends in the field, the challenges that remain to be addressed, and the potential for future research in this area.This review focuses on the aerial transportation of suspended-cable payloads using quadrotors. It provides an overview of the current state of the art in this field, covering the last sixteen years of research. The paper discusses various dynamic cable models and control systems used for both individual and collaborative transport scenarios. It highlights the challenges associated with modeling and controlling the nonlinear dynamics of quadrotor systems with suspended payloads, as well as the different strategies employed to address these challenges. The review also emphasizes the importance of cable modeling in achieving accurate and stable flight performance, and the need for robust control strategies to handle the complexities introduced by the suspended load. The paper discusses the advantages and limitations of different cable models, including taut-cable, elastic-cable, and flexible-cable models, and their respective applications in both individual and collaborative transport scenarios. It also explores the various control strategies used in the field, including anti-swing control, optimal trajectory following, and control for aggressive maneuvers. The review concludes with a discussion of the main trends in the field, the challenges that remain to be addressed, and the potential for future research in this area.