The twisting of a bar about its own axis without bending or extension is called pure torsion. Like the axial bar and beam problems, the theory of torsion is built on a kinematic hypothesis that results in a relationship between rotation and strain. This kinematic relationship combines with equilibrium and the constitutive behavior of the material to form a complete mathematical model of torsion. In this chapter, we formulate the theory of pure torsion and show how to solve the resulting equations. We start with bars with circular cross sections, for which the theory is exact. In the latter part of the chapter, we formulate approximate theories for a few important cases of bars that have non-circular cross sections.