It should be clear from the derivation of mesoscale bounds for linear elastic materials that three properties are required for their derivation: (1) the statistical homogeneity and ergodicity; (2) the Hill condition leading to admissible boundary conditions; and (3) an extremum (or variational) principle. It follows that the mesoscale bounds can be generalized to various other types of material behavior. More specifically, in this chapter we give hierarchies of mesoscale bounds for nonlinear elasticity, elastoplasticity, rigidplasticity, elastic-brittle damage, viscoelasticity, flow in porous media and thermoelasticity.