Continuum robots can traverse anatomical pathways to intervene at 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 are 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. Nowadays CTRs are being evaluated as tools for a variety of surgical applications as they can offer precision and manipulability in tight workspaces. This manuscript delivers an exhaustive classification of CTR research based on their clinical application, while also highlighting the explored approaches on modelling, control, design, and sensing. Competing approaches are critically presented, leading to a discussion on future directions to address current research and research translation limitations.