Herein, recent advances, current limitations, and open challenges in the design, modeling, and control of continuum robots are discussed. Thanks to their lean bodies, these robots achieve a long reach through narrow and tortuous environments, enabling currently unachievable tasks for medical, industrial, and service applications. The recent surge in research on these robots leads to significant advances in design, modeling, and control methods. Herein, these developments with a comprehensive review of existing continuum robots and emerging technologies are examined. Then, modeling and control approaches are compared, and navigation strategies exclusive to continuum robots, such as follow the leader, coiling, and circumnutation are examined. Finally, an overview of their applications is provided with a focus on industrial and service tasks, outlining open research challenges and future developments.