Modern high-energy particle accelerators and synchrotron light sources demand smaller and smaller beam emittances in order to achieve higher luminosity or better brightness. For light particles such as electrons and positrons, radiation damping is a natural and effective way to obtain low emittance beams. However, the quantum aspect of radiation introduces random noise into the damped beams, yielding equilibrium emittances which depend upon the design of a specific machine. In this dissertation, the author attempts to make a complete analysis of the process of radiation damping and quantum excitation in various accelerator systems, such as bending magnets, focusing channels and laser fields. Because radiation is formed over a finite time and emitted in quanta of discrete energies, he invokes the quantum mechanical approach whenever the quasiclassical picture of radiation is insufficient. He shows that radiation damping in a focusing system is fundamentally different from that in a bending system. Quantum excitation to the transverse dimensions is absent in a straight, continuous focusing channel, and is exponentially suppressed in a focusing-dominated ring. Thus, the transverse normalized emittances in such systems can in principle be damped to the Compton wavelength of the electron, limited only by the Heisenberg uncertainty principle. In addition,more » he investigates methods of rapid damping such as radiative laser cooling. He proposes a laser-electron storage ring (LESR) where the electron beam in a compact storage ring repetitively interacts with an intense laser pulse stored in an optical resonator. The laser-electron interaction gives rise to rapid cooling of electron beams and can be used to overcome the space charge effects encountered in a medium energy circular machine. Applications to the designs of low emittance damping rings and compact x-ray sources are also explored.« less