Definitions abound as the stage is set for the introduction and engagement of the terminology that underlies the foundation of hyperelasticity. The introduction of a strain-energy (or stored-energy) function into elasticity is due to George Green (1793–1841) and elastic solids for which such a function is assumed to exist are said to be Green elastic or hyperelastic. Elasticity without an underlying strain-energy function is called Cauchy elasticity. A formal definition is: hyperelasticity is the finite strain encompassing constitutive theory which describes the mechanical behavior of elastic solids with the use of one material function. The formulation of finite strain elasticity is considered with uncoupled, volumetric/deviatoric response and is based on the multiplicative decomposition of the deformation gradient. Additive decomposition, although formally valid, loses its physical content in the nonlinear theory. The volume-preserving, or isochoric, part of the deformation gradient is referred to as the distortion gradient, while the Jacobian determinant defines the volume change. Because the deformation gradient provides a complete description of homogeneous local deformations, it is considered to be the primitive measure of deformation.