In this chapter two major theories in continuum mechanics are briefly presented. The purpose of presenting these theories in the present book is to show two important applications of Tensor Analysis. Firstly, the theory of linearly elastic materials shows how the stress tensor T and the strain tensor E for small deformations are connected through constitutive equations , to provide a complete theory of wide applications. Secondly, the theory of linearly viscous fluids illustrates how the stress tensor T and the velocity field through the rate of deformation tensor D are combined through constitutive equations, to provide some important solutions to fluid mechanics problems. The constitutive equations of both theories are tensor equations and will be presented as such, with the component forms in Cartesian coordinates.