Transversely pumped dye-laser systems are investigated theoretically and experimentally. A set of coupled rate equations for the excited-state population densities and for the photon fluxes in both directions, at all wavelengths, is presented. Both the temporal and spatial dependence of these quantities are accounted for. The equations are solved numerically for a variety of practical situations, and analytical approximations for some limiting cases are discussed. The results describe the dependence of the amplified-spontaneous-emission (ASE) output flux on pumping rate, the spectral narrowing process, and the effects of gain saturation. It is found that under practical laboratory conditions the gain of such dye systems saturates rapidly. Consequently, at high pumping rates the output varies linearly with pump intensity, and the conversion efficiency from pump to ASE photons approaches unity. The performance of dye-laser amplifiers is described by the same set of equations, and the gain characteristics of such systems are analyzed as a function of input signal intensity and pumping rate. The theoretical calculations are compared with the results of a set of experiments, and good agreement is found. The operation characteristics of a dye-laser amplifier are evaluated and utilized in the design of a narrowband oscillator-amplifier system.