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In this work, problems and developments in the mechanics of slender, or rod-like bodies,are presented. We begin by offering a modern perspective on Green and Naghdi’s developments of the latter half of the 20th century for a directed rod. This review serves as background material to the more novel parts of this dissertation, which include applications and a discretization of the continuous theory. Governing equations for rods are developed by derivation from three-dimensional continuum mechanics and by direct approach. A treatment of constraints is also presented.After the background material is reviewed, we thoroughly describe a model for peristaltic locomotion using Green and Naghdi’s directed rod theory. The resulting model is applied to simulating motions of a compressible soft robot which uses Poisson’s effect for peristalsis. In addition, a calibration of parameters results in a validation of the model for use in biomimetic modeling of earthworm locomotion. Incompressibility of the worm is enforced as an internal constraint of the directed rod. In addition, a pair of muscle actuation models for a single continuum is included in our discussion.Finally, a discrete model for elastic rods undergoing planar motions is presented based on the theory of the directed rod. Discrete edge vectors and directors are used to capture cross section deformations including stretch, stretch gradients, shear, shear gradients, and the Poisson effect. In addition, deformations such as longitudinal stretch and bending are also incorporated. The model is validated with the help of known analytical solutions to benchmark problems from Green and Naghdi’s continuous rod theory.