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The mean state, variability, and extreme variability of the stratospheric polar vortices, with an emphasis on the Northern Hemisphere (NH) vortex, are examined using two-dimensional moment analysis and extreme value theory (EVT). The use of moments as an analysis tool gives rise to information about the vortex area, centroid latitude, aspect ratio, and kurtosis. The application of EVT to these moment-derived quantities allows the extreme variability of the vortex to be assessed. The data used for this study are 40-yr ECMWF Re-Analysis (ERA-40) potential vorticity fields on interpolated isentropic surfaces that range from 450 to 1450 K. Analyses show that the most extreme vortex variability occurs most commonly in late January and early February, consistent with when most planetary wave driving from the troposphere is observed. Composites around sudden stratospheric warming (SSW) events reveal that the moment diagnostics evolve in statistically different ways between vortex splitting events and vortex displacement events, in contrast to the traditional diagnostics. Histograms of the vortex diagnostics on the 850-K (~10 hPa) surface over the 1958--2001 period are fitted with parametric distributions and show that SSW events constitute the majority of data in the tails of the distributions. The distribution of each diagnostic is computed on various surfaces throughout the depth of the stratosphere; it shows that in general the vortex becomes more circular with higher filamentation at the upper levels. The Northern and Southern Hemisphere (SH) vortices are also compared through the analysis of their respective vortex diagnostics, confirming that the SH vortex is less variable and lacks extreme events compared to the NH vortex. Finally, extreme value theory is used to statistically model the vortex diagnostics and make inferences about the underlying dynamics of the polar vortices. [ABSTRACT FROM AUTHOR]