The theory of stimulated Thomson scattering is investigated both quantum mechanically and classically. In the interaction of a collisionless plasma with two electromagnetic waves, both with frequencies well above the plasma frequency, energy is transferred from the high-frequency wave to the low-frequency wave via stimulated Thomson scattering. This process is mediated by the nonlinear interaction of the plasma electrons with a beat wave at the difference frequency between the two waves; this beat frequency must be well above the plasma frequency. The gain coefficient for stimulated Thomson scattering is calculated both quantum mechanically and classically, and identical results are obtained. The classical calculation also yields the first nonlinear term in the index of refraction due to stimulated Thomson scattering, as well as the details of the saturation of the gain and the index of refraction. The authors present explicit formulas for the gain coefficient and the index of refraction, in the unsaturated limit, for both very cold and very hot plasmas. The calculations indicate that it should be possible to detect stimulated Thomson scattering experimentally by means of polarization enhancement. In an appendix, the theory of stimulated Thomson scattering is used to treat the free-electron amplifier.