Continuum robots can traverse anatomical pathways to intervene in regions deep inside the human body. They are able to steer along 3D curves in confined spaces and dexterously handle tissues. Concentric tube robots (CTRs) are continuum robots that comprise a series of precurved elastic tubes that can be translated and rotated with respect to each other to control the shape of the robot and tip pose. CTRs are a rapidly maturing technology that has seen extensive research over the past decade. Today, they are being evaluated as tools for a variety of surgical applications, as they can offer precision and manipulability in tight workspaces. This review provides an exhaustive classification of research on CTRs based on their clinical applications and highlights approaches for modeling, control, design, and sensing. Competing approaches are critically presented, leading to a discussion of future directions to address the limitations of current research and its translation to clinical applications. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 5 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.