This thesis has focused on analyzing the coupled axial-torsional dynamics of a drilling structure, due to its importance within the drilling activity. This work builds upon earlier published work on the coupled axial-torsional dynamics of drilling systems, which addressed the problem by means of either low-order or nite element models. During the course of this study we realized that the spatial discrete representation of the drilling system plays an important role in the model stability properties; in particular the model tends to become more unstable when it is represented by a larger number of DOF's (i.e. a ner discretization). Ultimately, such a lack of stability can be an inherent property of the drilling system. If the observed instability of the drilling system is indeed an inherent property, only signicant changes to the dynamics can provide system stability. Focusing on that aspect, a nite element model was used to investigate the value of the use of a simple (industrial) angular velocity drive system control (Soft Torque) to mitigate stick-slip. The Soft Torque controller can be represented by a spring-dash pot surface boundary condition which is tuned to damp the rst torsional natural frequency of the drill string. From the results in the thesis, it can be concluded that the coupled axial-torsional dynamics, with the bit/rock interface, cannot in general be stabilized by the Soft Torque controller. This is likely to be related to the fact that higher modes of the drill-string dynamics play a role in instabilities leading to stick-slip oscillations. Motivated by this observation, this study took on the challenge of investigating which level of discretization provides and accurate description in the dynamics. To understand the role of spatial discretization of the drill-string dynamics, discrete models were developed to understand stability properties and to study the overall time-domain behavior. Based on a time scale separation argument (between the axial and torsional dynamics), 1-DOF, 2-DOF, and multi-DOF lumped parameter models describing only the axial dynamics of the drillstring were studied. Subsequently, the coupled dynamics of one and two identical oscillators were investigated. In all cases, the increasing number of oscillators led to a more unstable system.