We systematically analyse all (110) EXOSAT AGN medium-energy light curves of length greater than 20 ks to search for and quantify their flux variability. We also perform extensive simulations to quantify the selection effects (such as the source brightness or the length of observation) that might prevent the detection of variability. We quantify variability in terms of a 'normalized variability amplitude' (NVA) derived from the power spectrum of each light curve. Of the 32 sources observed, we detect variability in 12. We find that, contrary to some previous claims, there is no evidence that sources change the character of their variability between observations. We confirm that there is a strong inverse correlation between NVA and source luminosity which can be simply explained if source size scales with luminosity. A further result is that the sources with the steepest energy spectra are possibly the most variable. One explanation of the latter result, in the context of reprocessing models, might be that sources with steeper energy spectra contain a smaller reflected or reprocessed component, which might be expected to vary less rapidly than the direct component. We briefly discuss a small number of light curves obtained using the low-energy telescope on EXOSAT, and confirm that their power spectra are steeper than those of the corresponding medium-energy light curves. The power spectra are consistent with both multiple decay shot-noise models and with the rotating-spot models. In the case of the rotating-spot model we can place upper limits on the central black hole masses of typically a few x 10(7) M.. Starburst models cannot explain the observed rapid X-ray variability.