In the literature, there are various material models proposed so as to model the constitutive behavior of hyperelastic materials for example, St. Venant-Kirchho_ model, Mooney-Rivlin model etc. The stability of such material models under various states of deformation is of important concern, and generally stability analysis is conducted in homogeneous states of deformation. Within hyperelasticity, instabilities can be broadly classified as geometrical and material types. Geometrical instabilities such as buckling, symmetric bifurcation etc. are of physical origin, and lead to multiple solutions at critical stretch. Material instability is a aw in the material model and leads to unphysical solutions at the onset. It is required that the constitutive model should be materially stable i.e., should not give unphysical results, and be able to predict correctly the onset of geometrical instabilities. Certain constitutive restrictions proposed in the literature are inadequate to characterize such instabilities. In the work, we propose stability criteria which will characterize geometrical as well as material instabilities. A new elasticity tensor is defined, which is found to characterize material instability adequately. In order to investigate the validity of proposed stability criteria, three important constitutive models of hyperelasticity viz., St. Venant-Kirchho_, compressible Mooney-Rivlin and compressible Ogden models are investigated for stability.