The description of linear random waves in the form of an amplitude-modulated carrier wave is known as narrow-band representation. Herewith, the theoretical basis of such a representation is examined in terms of integral properties of surface spectra and criteria governing the statistical and kinematic characteristics of the carrier wave. These considerations are then extended systematically to derive two narrow-band type representations for nonlinear waves. The nature of these representations and their statistical properties are discussed and compared with other models such as those proposed earlier by Tayfun (1980) and Huang et al. (1983).