To study the behavior of an elastic continuous medium, the theory of elasticity is a generally accepted model. A simple idealized linear stress-strain relationship gives a good description of the mechanical properties of many elastic materials around us. By this relation, we need 21 elastic constants to describe a linear anisotropic elastic material if the materials do not possess any symmetry properties. Consideration of the material symmetry may reduce the number of elastic constants. If the two-dimensional deformation is considered, the number of elastic constants used in the theory of elasticity can be further reduced. If the materials are under thermal environment, additional thermal properties are needed to express the temperature effects on the stress-strain relation. If the materials exhibit the piezoelectric effects, the stress-strain relation should be further expanded to include the electric displacements and the electric fields. If not only the inplane deformation but also the out-of-plane deflection are considered for the laminates made by laying up various unidirectional fiber-reinforced composites, the elastic constants will generally be reorganized into the extensional, coupling and bending stiffnesses to suit for the classical lamination theory. Since the computer program developed in this book covers all these kinds of materials, their constitutive relations are now described in this Chapter. Further extensions to magneto-electro-elastic and viscoelastic materials will then be described in Chaps. 11 and 12.