This chapter presents theoretical and numerical formulations of nonlinear elastic materials. Since nonlinear elastic materials typically experience a large deformation, Sect. 3.2 discusses stress and strain measures under large deformation. Section 3.3 shows two different formulations for representing large deformation problems: total Lagrangian and updated Lagrangian. In particular, it is demonstrated that these two formulations are mathematically identical but differ in computer implementation and the interpretation of material behaviors. Critical load analysis is introduced in Sect. 3.4, followed by hyperelastic materials in Sect. 3.5. Different ways of representing the incompressibility of elastic materials are discussed. The continuum form of the nonlinear variational equation is discretized in Sect. 3.6, followed by a MATLAB code for a hyperelastic material model in Sect. 3.7. Section 3.8 summarizes the usage of commercial finite element analysis programs to solve nonlinear elastic problems, particularly for hyperelastic materials. In hyperelastic materials, it is essential to identify material parameters. Section 3.9 presents curve-fitting methods to identify hyperelastic material parameters using test data.