Linear shear properties of human and bovine brain tissue were determined from transient stress-relaxation experiments and their material functions were compared. Quasi-linear viscoelastic theory was then utilized to determine material constants for bovine brain tissue subjected to large deformations. The range of applicability for linear and quasi-linear constitutive models of brain tissue was determined. A nonlinear Green-Rivlin constitutive model was subsequently applied to characterize temporal nonlinearity of bovine brain tissue in shear. Overall, 10 brain specimens from 5 fresh human cadavers and 156 brain specimens from 26 fresh bovine cadaver brains were used to quantify and compare shear brain responses under various loading conditions. The assumptions of homogeneity, isotropy, and incompressibility of brain material were made in order to reduce the required number of experiments. A series of single-, two-, and three-step strain inputs was applied to one end of a cylindrical brain specimen and the stress-time histories were measured at the other end. The time delays between the applied strain step inputs were altered in order to determine the temporal nonlinearity of brain tissue. The study resulted in linear constitutive models for human and bovine brain tissue, and quasi-linear and nonlinear constitutive equations for bovine brain tissue in shear. It was found that human brain is somewhat stiffer than bovine brain; the difference, however, was not statistically significant and bovine brain may be a good substitute in studying nonlinear human brain response. A linear constitutive model was found to be sufficient to characterize brain tissue response when Lagrangian shear strains do not exceed 0.2 (advised to limit the range to the shear strain of 0.175), a quasi-linear constitutive model can then be used for loading conditions of up to 0.5 of Lagrangian shear strains (advised to limit the range to the shear strain of 0.325). For any shear strain magnitudes and histories a fully nonlinear Green-Rivlin viscoelastic constitutive model may be utilized.