In this chapter we present constitutive relations for three types of neo-Hookean isotropic hyperelastic material models: incompressible (classical) model (Sect. 3.1), compressible mixed models (Sect. 3.2), and compressible vol-iso models (Sect. 3.3). The formulations of these three types of models are discussed in Sect. 3.4. Although all three types of models are presented in one form or another in the literature (see Chap. 1), the objectives of the present chapter is to unify them in the Eulerian formulation and relate them to the linear isotropic elastic models formulated in Sect. 2.4.