Medical robotics is, since roughly 40 years, a continuously flourishing field that combines the advances in robotics and healthcare to provide improved patient outcomes. Today, most robots used in clinic are rigid-link robots, characterized with mechanical rigidity and limited degrees-of-freedom. In recent years, however, there has been a growing trend towards the development of robots that are designed to interact with the human body in a more compliant manner and offer increased dexterity to the clinicians. Moreover, to further reduce surgical invasiveness, a search has started for ways to access the body through natural orifices. This has led to the advent of continuum robots in the medical robotics field. Continuum robots are composed of elastic materials such that their structure bends continuously when actuated, providing infinite degrees of freedom. Such degrees of freedom give this type of robots the ability to conform to lumens of the human anatomy and access regions that are difficult to reach with traditional robots. In many cases, continuum robots can be scaled down to very small sizes (0.5 mm in diameter), yet further opening the range of possible applications. Additionally, their compliant nature make them especially suited for safe interactions in medical applications. While continuum robots clearly offer new advantages, this emerging type of robots also presents a series of new challenges to the community. First, structurally, such robots can be subject to a number of physical phenomenons that limit their performances. Second, in order to deploy continuum robots, precise models that link actuation variables to the full robot shape are required. In this second challenge, the modeling of continuum robots is drastically different from that of rigid-link robots. The elasticity of the constituting materials must be taken into account, along with interaction forces from the environment. To answer these challenges, the continuum robotics community is continuously developing novel robot structures and hence new models. In this dissertation, the design and modeling of a novel continuum robot structure, namely concentric agonist-antagonist robot (CAAR), has been investigated. CAARs are identified as promising robots and their potential for medical applications is analyzed. In order to deploy CAARs and control their shape or tip position, a precise mechanical model was yet to be derived. Models are evidently used for control purposes but also for design optimization, path planning, feedback, and the analysis of their stability, manipulability etc. The more precise and capable continuum robot models are, the more advantageously they can be exploited for these various tasks. This thesis proposes such a (geometrically exact) mechanical model based on the Cosserat rod theory. The model is derived through a Lagrangian approach, which appeared more recently in the community than the better known Newtonian approach that was initially developed. While the Newtonian approach isolates robot constituents and considers their interactions (Newton's Laws of motion), the Lagrangian approach considers the system as a whole and derives the robot models through a canonical application of the principle of virtual work. By considering the system in its entirety, this approach promises to provide a more comprehensive understanding of the behavior of continuum robots. The similarities and differences between both approaches are further discussed in this thesis, including a side-by-side comparison in a case study with tendon actuated continuum robots. It is this in depth analysis that enabled to derive a Cosserat rod model for CAARs. Finally, the derived model is validated through an extensive set of experiments using various 3D-printed CAAR designs, in free-space and with external forces applied at the robot tip. Results show that the model performs well with mean tip error of 1.47% of the robot length, over all experiments.