This thesis identifies, develops and applies methods for the decomposition of fading dispersive channels. Such channels arise in wireless communication as a result of multipath and relative motion of the transmitter, scatterers and receiver. The decompositions considered are the f-power series and Karhunen-Loève (KL) expansions. For the KL expansion generalisations to rapid time variation are possible with the separate options of single spread and double spread decomposition. The single spread decomposition involves a model of the instantaneous channel transfer function with time variation supported by sample spaced coefficients. The double spread decomposition employs a model of each received pulse and requires symbol spaced coefficients. The decompositions are applied to pulse shaping, channel modelling for equalisation and the determination of performance limits for linear modulation over fading dispersive channels. The results on pulse shaping show that, with moderate bandwidth expansion and appropriate design, it is possible to significantly lower the complexity of a mobile receiver. The approach suggests a way to move complexity and power consumption away from the mobile unit and into the base station. The effects of diversity on performance are investigated by assuming a single pulse from a linear modulation format. This removes the need to consider intersymbol interference and allows conclusions about the impact of fading and dispersion on the probability of error and the average mutual information.