Hyperelastic materials exhibit excellent mechanical properties and have been widely applied in engineering. To accurately calculate the large deformation response of hyperelastic materials, this study reviews eight commonly used macroscopic hyperelastic constitutive models and presents fitting methods. The stress-strain relationships for Mooney-Rivlin, Neo-Hookean, Yeoh, Ogden, Arruda-Boyce, Van der Waals, and HyperFoam models under uniaxial tension, equibiaxial tension, pure shear, and volumetric compression are derived. Linear least squares and Levenberg-Marquardt methods are employed to fit linear and nonlinear constitutive models, respectively. The validity of the constitutive models and fitting methods is verified through comparison with experimental data from uniaxial tensile tests.