In this chapter, a number of concepts of the solid mechanics are introduced to describe a linear elastic body in terms of the partial differential equations. In particular, the displacement vector, strain tensor, and stress tensor fields are introduced to define a linear elastic body which satisfies the strain-displacement relations, the equations of motion, and the constitutive relations. Emphasis is placed on the compatibility relations, the general solutions of elastostatics, and on an alternative definition of the displacement field of elastodynamics. A discussion of the constitutive relations includes the orthotropic, transversely isotropic, and isotropic materials. The stored energy of an elastic body, the positive definiteness and strong ellipticity of the elasticity fourth-order tensor, and the stress-straintemperature relations for a thermoelastic body are also discussed. The chapter contains a number of worked examples, and also end-of-chapter problems with solutions provided in the Solutions Manual.