This thesis is a report of work which develops the study of electromagnetic radiation by accelerating charges in the scalar quantum electrodynamic theory. We investigate aspects of this theory in flat spacetime, and in a class of conformally flat and curved spacetimes. In particular, we show that in flat spacetime, the quantum-theoretic prediction for the emission of energy by the particle, in the limit h-bar tends to 0 and to order e^2 in the coupling constant, may be shown to match the classical calculation. We also calculate the order h-bar correction to this quantity for two specific classes of problem. In the class of conformally flat and curved spacetimes, we compare the change in position due to the radiation reaction with the classical result, and we also consider some of the one-loop corrections to the theory. We show that as h-bar tends to 0, the conformally flat result and the classical result match, but that in that limit the general spacetime results differ.