Papers dealing with the generalized Hooke's law for linearly elastic anisotropic media are reviewed. The papers considered are based on Kelvin's approach disclosing the structure of the generalized Hooke's law, which is determined by six eigenmoduli of elasticity and six orthogonal eigenstates.Many natural materials, such as rocks, crystals, and biological tissues, and also materials used in advanced technologies, in particular, composites, are characterized by substantial anisotropy of their elasticity properties. In most cases, composite materials, as well as their components, are anisotropic materials. To create composite materials with elasticity properties necessary for engineering practice, one should know admissible limits of the components of the tensor of the elasticity moduli and the tensor of the compliance coefficients of anisotropic materials.The constitutive equations of the linear theory of elasticity [1][2][3][4][5] in the Cartesian rectangular coordinate system (x 1 , x 2 , x 3 ) include the equations of motionthe generalized Hooke's lawand the Cauchy formulas, which express strains via displacements:In Eqs.(1)-(3), σ ij = σ ji are the components of the symmetric stress tensor, ε ij = ε ji are the components of the strain tensor, E ijkl are the components of the fourth-rank tensor of the elasticity moduli, u i are the components of the displacement vector, F i are the components of the vector of bulk forces, ρ is the constant density of the material, and t is the time. The comma ahead of the subscript indicates differentiation with respect to the spatial coordinate marked by this subscript; repeated letters in the subscripts indicate summation over their admissible values. Relations (2) can be inverted: ε ij = S ijkl σ kl (S ijkl are the components of the fourth-rank tensor of the compliance coefficients).In the linear theory of elasticity, the specific strain energy for anisotropic materials is presented as [1, 2]The components E ijkl possess the properties of symmetry [2, 5]:The constants S ijkl also satisfy the conditions of symmetry (5) and are related to E ijkl by the expressions E ijkl S klrs = δ ijrs ≡ (δ ir δ js + δ is δ jr )/2, S ijkl E klrs = δ ijrs ,