Coherent electron cooling (CeC) offers the potential a very potent method of longitudinal phase-space cooling for high intensity bunched beam accelerators, such as at the Relativistic Heavy Ion Collider (RHIC) or at proposed electron-ion colliders such as eRHIC or LHeC. To develop a complete theoretical description of CeC requires a detailed model of the phase space dynamics of a high-gain free-electron laser (FEL) in three dimensions. A three-dimensional model for the FEL instability is developed using the Maxwell-Vlasov formalism, and obtains a Green function for arbitrary initial phase space perturbations. This Green function assumes a transversely infinite electron beam with zero transverse velocity spread. The formalism developed for obtaining the Green function also provides a solution to the initial value problem of an FEL with a finite transverse beam, and this formalism is used to obtain optical guiding. Using the resulting dispersion relation for the FEL process, I present a number of theorems and results concerning the roots of the dispersion relation, in particular that regardless of the specific functional form of the thermal background of the beam there is one and only one amplifying mode. A number of criterion and relations on that mode is also developed and presented. Finally, I develop a theoretical description of the dynamics of Coherent Electron Cooling considering the case of a finite length electron bunch which paints the longer hadron bunch. This leads to a kinetic equation for the cooling of synchrotron oscillations in bunched beams.