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The interaction of relativistic electrons and radiation is studied theoretically and with computer simulations for the amplification of radiation or for the acceleration of electrons. For radiation amplification, we investigate the following two mechanisms: 1.) AC Free Electron Laser (ACFEL) in which a relativistic electron beam (with relativistic factor ɣo is wiggled by an AC (temporally oscillating but spatially uniform) transverse electric or magnetic structure. Radiation is produced at frequency f~2ɣo2 fo (for ɣo >>1), where fo is the frequency of the AC field. A rectangular wave-guide design is presented to illustrate one realization of the ACFEL and its performance is compared to conventional FEL's; 2.) Cherenkov Maser in which a mildly relativistic electron beam is propagated along the axis of a dielectric-lined wave-guide so as to produce microwaves. A newly developed particle simulation code is used to model the TH mode coupling in this scheme. Good agreement is obtained between the simulations and a recent laboratory experiment.In both cases (1 and 2), the nonlinear saturation is found to be caused by the trapping of the electrons in self generated longitudinal waves. Momentum spread in the electron beam (beam temperature) is observed to result in smaller gain per unit length and reduced saturation efficiency. It is also demonstrated by our computer simulations that the output power can be enhanced if an axial dc electric field of appropriate magnitude is imposed immediately before saturation.For electron acceleration, we consider laser-plasma accelerator schemes. In these schemes, electromagnetic radiation (e.g. a laser) drives a plasma space charge wave which can trap and accelerate relativistic electrons at rate of order 1 Gev/cm (for a plasma of density 1018cm-3). In particular, we propose and investigate a new scheme to generate the plasma waves by a monochromatic laser injected from the side rather than colinearly with the accelerated electrons. This overcomes the pump depletion problem inherent in previous colinear schemes (e.g. Beat-Wave Accelerators).