A rich variety of dynamical processes underlie the operation of active semiconductor light-emitting devices, such as semiconductor optical amplifiers. These processes include interband and intraband carrier dynamics. Interband processes comprise spontaneous recombination, both radiative and Auger, stimulated radiative recombination, and carrier transport. Intraband processes comprise carrier heating and cooling and spectral hole-burning, among others. The dynamical processes affect both the gain and refractive index of the semiconductor optical amplifier. In this article, these dynamic processes and their physical origins are reviewed. Under conditions of large, time-varying changes in carrier density or intraband carrier distribution, nonlinear gain and refraction becomes significant. For applications requiring linear amplification, such nonlinearities are deleterious. However, for many applications such nonlinearities can be the basis for useful device functions. In particular, the nonlinearities of cross-gain modulation, cross-phase modulation, and four-wave mixing in semiconductor optical amplifiers have been applied for the functions of wavelength conversion, optical time-demultiplexing, clock recovery, and trans-multiplexing. Such nonlinear devices based on semiconductor optical amplifiers and their effects on propagating optical signals are also reviewed.