Torsional vibration of the drill strings used in drilling oil and gas wells arises from a complex interaction of the dynamics of the drilling structure with speed-dependent effective rock-cutting forces. These forces are often difficult to model, and contribute substantially to the problems of controlling the drilling operation so as to produce steady cutting. We consider here the use of a nonlinear energy sink (NES), an attachment which has been shown to be effective in reducing or even eliminating self-excited motions in van der Pol and aeroelastic systems. The NES is a completely passive, inherently broadband vibration absorber capable of attracting and dissipating vibrational energy from primary structures, in this case nonlinear discontinuous models of drill-string systems. In this paper we describe a prototypical drill string-NES system, briefly discuss some of the analytical and computational tools suitable for its analysis, and then concentrate on mathematical results on the efficacy of the NES in this application and their physical interpretation.