The work presented in this thesis is a study of the X-ray variability of Active Galactic Nuclei (AGN) as observed with three recent X-ray astronomy satellites, EXOSAT, Ginga and ROSAT. The nature of the variability is investigated and then compared with the most recent models of X-ray variability in AGN. The occurrence, and nature, of rapid X-ray variability in AGN has been studied using data from the EXOSAT database. The precise nature of the varibility has been investigated by means of the power spectrum of the light curve. The relationship between the variability amplitude and the luminosity of the source has been investigated, and it has been found that the variability amplitude is approximately α L_x^-1/3, where L_x is the X-ray luminosity of the source. The variability amplitude also appears to be related to the energy spectrum of the source. A small number of Ginga light curves have been analysed, and the trends found using the EXOSAT data are confirmed. The data also reject the hypothesis that there is a common power spectrum slope for all AGN. There is a strong inverse correlation between the amount of reflected emission observed in the energy spectrum of the source, and the variability amplitude. The low luminosity Seyfert I galaxy, NGC 4051 has been observed with the ROSAT observatory, and the results of the analysis of these observations are presented. The flux variability is found to be non-stationary with the power spectrum having a different slope at different epochs. The possibility of the X-ray variability being a chaotic process is raised. The energy spectrum is well fitted by the warm absorber model. NGC 4051 shows significance spectral variability, which can also be explained in terms of the warm absorber model. The overall data favour the complex shot noise model as the X-ray production mechanism, with the shots possibly being caused by magnetic recombination on the surface of an accretion disc. The X-ray emission undergoes extensive reprocessing via both reflection from cold material, probably the accretion disc, and absorption by warm material in the line of sight.