This chapter describes the modeling of mechanical systems in order to describe their nonlinear vibratory response, focusing on free vibration and frequency response near resonance. Some general thoughts about modeling are offered first, along with some archetypical models for nonlinear vibration, motivated by physical examples. We focus on considerations related to the inclusion of nonlinearities in system models and how nonlinearity affects the ability of a model to describe system response. Methods for analyzing nonlinear vibration systems are not emphasized, and only the minimum tools required to demonstrate the results of interest are introduced. Models with one and two modes with polynomial nonlinearities are considered, since these offer descriptions of generic behaviors that are quite common and qualitatively distinct from linear system response. Sample physical systems that are used to demonstrate the main ideas include the simple pendulum, transverse vibrations of beams, capacitively driven micro-systems, the spherical pendulum, and a spring-pendulum system. The chapter closes with some general thoughts about nonlinear system modeling and its use in design.