Nonlinear energy sinks (NES) have been shown to be effective for vibration attenuation under certain conditions. The effects of coupling linear systems to NES with nonlinear stiffness and linear damping have been extensively investigated. However, research involving nonlinearly damped NES is minimal. In this thesis, the performance of nonlinearly damped NES are examined and comparisons are made with the performance of linearly damped counterparts. Saddle-node and Hopf bifurcation diagrams are presented to show instabilities of solutions. The strongly modulated response (SMR), which exists only near 1:1 resonance between the forcing frequency and the frequency of the linear system, is investigated as well. Finally, time response comparisons are made between the systems with linearly and nonlinearly damped NES.