We have studied the climatological structure, interannual variability and interrelationship of various characteristics of the Arctic and Antarctic stratospheric vortices at several isentropic levels on the basis of the ECMWF ERA-40 reanalysis (1957--2002). Because of suspect data in the presatellite period in the Southern Hemisphere, only data from 1979 onward are used to study the climatology of the Antarctic vortex. The climatological structure of the vortices in both hemispheres is mainly consistent with previous climatologies obtained from other data sets. A study of the interrelationship between the vortex characteristics suggests that in the Arctic a larger vortex is usually colder and stronger whereas in the Antarctic winter such a relationship is not established. It is found that the Arctic PSC area has increased during the 1958--2002 period, but, in contrast to earlier studies, no statistically significant trends in size, coldness or longevity of the Arctic lower-stratospheric vortex since 1979 are found. During the period 1979--2001 the Antarctic spring vortex has become stronger and colder, and it breaks up later. However, the Antarctic vortex cooling has not affected the October vortex area, which shows only little change for the same period. It is found that the area of the Antarctic vortex during late winter and spring depends on the planetary wave propagation to the stratosphere in the preceding period, whereas the corresponding relationship between these waves and the PSC area in October is destroyed by the trends in the PSC area.