Aims. Using archival observations recorded over a 5+ year timeframe with the NASA GALaxy Evolution eXplorer (GALEX) satellite, we present a study of the ultraviolet (UV) variability of 4360 quasars of redshifts up to z = 2.5 that have optical counterparts in the Sloan Digital Sky Survey DR5 spectroscopic catalog of Schneider et al. (2007, AJ, 134, 102). The observed changes in both the far UV (FUV: 1350−1785 Å) and near UV (NUV: 1770−2830 Å) AB magnitudes as a function of time may help differentiate between models of the emission mechanisms thought to operate in these active galaxies. Methods. A list of NUV and FUV variable quasars was derived from the UV light-curves of sources with 5 or more observational visits by GALEX that spanned a time-frame >3 months. By measuring the error in the derived mean UV magnitude from the series of GALEX observations for each source, quasars whose UV variability was greater than the 3-σ variance from the mean observed value were deemed to be (intrinsically) UV variable. This conservative selection criterion (which was applied to both FUV and NUV observations) resulted in identifying 550 NUV and 371 FUV quasars as being statistically significant UV variable objects. Results. Following the work of Vanden Berk et al. (2004, ApJ, 601, 692), we have performed a structure function (SF) analysis of these data to search for possible correlations between UV variability and parameters such as rest frame time-lag, redshift, luminosity and radio loudness. Firstly, we observe that the amplitudes of variability as a function of time-lag for both the NUV and FUV data are far larger than those observed at visible wavelengths. Secondly, the levels of FUV variability are greater than those observed in the NUV for a given value of time-lag. Also, the amplitudes of both the NUV and FUV variability of quasars increase as a function of rest frame time-lag, irrespective of the value of quasar redshift, for time-lags <200 days. For time-lags >300 days there is a pronounced rollover in the NUV SF for all redshift values, which is also observed (with a lower signiÞcance) in the FUV variability data. Although we find no strong relationship between UV variability and redshift, our data do show that higher redshift quasars appear to be more variable than their low redshift counterparts. Our data also show that, for all values of time-lag, the more luminous quasars tend to be slightly less UV variable, with perhaps the exception of FUV variable quasars for short time-lags. Although our data sample is small, we find that radio-loud quasars are marginally more variable than radio quiet ones by a factor ∼2 in the NUV and by a factor 1−3 in the FUV. Therefore, our present observations support the notion in which the radio properties of quasars have a limited influence on the observed UV variability of these objects. In summation, our present analysis favors a quasar model in which UV variability is mainly due to stochastic changes in the underlying continuum level, rather than models that favor gravitat...