This paper reviews the current state of aerial transportation systems using suspended loads by single and multiple quadrotors, focusing on dynamic cable models and control systems. The review covers the past sixteen years of literature and discusses the main trends in research. The authors highlight the challenges and advancements in cable modeling, including taut-cable, hybrid, and elastic cable models, and control strategies for both individual and collaborative transport. The paper also addresses the limitations of current models, such as the simplifications made in dynamic equations and the computational costs associated with advanced control methods. The authors conclude by identifying future challenges, such as the need for more realistic outdoor experiments, the development of testing platforms for vehicles carrying suspended loads, and the exploration of learning-based systems for robustness in transportation tasks. The review aims to provide a comprehensive analysis and valuable insights into the key facets of aerial transportation systems with suspended loads by quadrotors.This paper reviews the current state of aerial transportation systems using suspended loads by single and multiple quadrotors, focusing on dynamic cable models and control systems. The review covers the past sixteen years of literature and discusses the main trends in research. The authors highlight the challenges and advancements in cable modeling, including taut-cable, hybrid, and elastic cable models, and control strategies for both individual and collaborative transport. The paper also addresses the limitations of current models, such as the simplifications made in dynamic equations and the computational costs associated with advanced control methods. The authors conclude by identifying future challenges, such as the need for more realistic outdoor experiments, the development of testing platforms for vehicles carrying suspended loads, and the exploration of learning-based systems for robustness in transportation tasks. The review aims to provide a comprehensive analysis and valuable insights into the key facets of aerial transportation systems with suspended loads by quadrotors.