The theory of elasticity deals with the systematic study of the stress, strain, and displacement fields in an elastic body under the influence of external forces. In this chapter, we deal with elastic materials (i.e., materials in which the deformation and stress disappear with the removal of the external forces). However, equations for other types of materials are also given. Index notation and Cartesian tensors are presented to enable equations of a continuum to be displayed with remarkable brevity. The concept of stresses and the deformation of a medium are explained. The constitutive or material laws, characterizing the mechanical behavior of the medium, are given. The application of the basic principles of mechanics to the deformable body is described. The elastic properties of biological materials along with examples of selected materials are presented. Also, biomechanical methods for measuring the elastic constants of biological materials as well as applications to diagnosis and treatment are demonstrated. Finally, the problem of free vibrations of an elastic cavity, simulating the human skull, and its solution is given.