Nanomechanical resonators made from graphene are widely researched for their potential applications due to their high sensitivity. Their atomic scale thickness makes it that these resonators exhibit amplitude-dependent damping at relatively small driving forces. To further increase the performance of these devices, it is essential to understand the dissipation process. Although many physical mechanisms that lead to linear damping have been investigated, the origin of nonlinear damping remains largely unknown. Nonlinear damping in nanomechanical resonators is typically studied using phenomenological models. In this thesis, a multilayer graphene resonator is studied that is electrostatically actuated. The graphene resonator is modeled as a viscoelastic membrane operating in the geometrically nonlinear regime. The amplitude-dependent damping term used in phenomenological models arises naturally in the equation of motion. The experimentally obtained nonlinear frequency responses are then fitted using a fully automated algorithm. Based on these fits, the viscoelastic properties of graphene and its loss tangent are found. When the same analysis is done for a single-layer graphene resonator coupled to an optical cavity, negative nonlinear damping is observed which turns positive with increasing laser power. The effect of the optical field on the nonlinear damping is then modeled. It is found that the change in equilibrium position of the graphene membrane due to laser illumination as well as geometric imperfections can lie at the root of the negative nonlinear damping behavior.