Representations in complete and irreducible forms for tensor functions allow general consistent invariant forms of the nonlinear constitutive equations and specify the number and type of the scalar variables involved. They have proved to be even more pertinent in attempts to model mechanical behavior of anisotropic materials, since here invariant conditions predominate and the number and type of independent scalar variables cannot be found by simple arguments. In the last few years, the theory of representations for tensor functions has been well established, including three fundamental principles, a number of essential theorems and a large amount of complete and irreducible representations for both isotropic and anisotropic tensor functions in three- and two-dimensional physical spaces. The objective of the present monograph is to summarize and recapitulate the up-to-date developments and results in the theory of representations for tensor functions for the convenience of further applications in contemporary applied mechanics. Some general topics on unified invariant formulation of constitutive laws are investigated.