In 1855 Fick reported on the diffusion of liquid through a membrane and proposed that there was an analogy between this process and that of heat conduction allowing him to transcribe the mathematical equation for heat conduction derived in 1822 by Fourier into a form to represent this diffusion of liquid. This model, known as Fickian diffusion, has become the baseline against which the characteristics of liquid diffusion are measured to the point where anomalous diffusion is known generically as non-Fickian. Numerous authors have attempted to develop models to cover all aspects of non-Fickian diffusion resulting in a very large number of models that consider the effect of parameters as varied as the chemical makeup, geometric dimensions, environmental history, stress state, and damage status of the material, as well as the likelihood of multiple diffusion mechanisms being responsible for transport of the water molecules. Of particular interest to structural engineers is the transport of moisture in polymer matrix composites owing to the plasticizing effect the moisture may have on the composite and the potential for the moisture to induce localized damage. This paper reviews analytical models that are relevant to the transport of moisture in structural composites. In doing so the benefits and limitations of the various models and techniques are presented in order to provide a reference for scientists and engineers attempting to describe the kinetics of moisture in composites accurately. There are 160 references cited in this review article.