  We use the data from Gaia Early Data Release 3 (EDR3) to study the kinematic
properties of Milky Way globular clusters. We measure the mean parallaxes and
proper motions (PM) for 170 clusters, determine the PM dispersion profiles for
more than 100 clusters, uncover rotation signatures in more than 20 objects,
and find evidence for radial or tangential PM anisotropy in a dozen richest
clusters. At the same time, we use the selection of cluster members to explore
the reliability and limitations of the Gaia catalogue itself. We find that the
formal uncertainties on parallax and PM are underestimated by 10-20% in dense
central regions even for stars that pass numerous quality filters. We explore
the the spatial covariance function of systematic errors, and determine a lower
limit on the uncertainty of average parallaxes and PM at the level 0.01 mas and
0.025 mas/yr, respectively. Finally, a comparison of mean parallaxes of
clusters with distances from various literature sources suggests that the
parallaxes (after applying the zero-point correction suggested by Lindegren et
al. 2021) are overestimated by 0.01+-0.003 mas. Despite these caveats, the
quality of Gaia astrometry has been significantly improved in EDR3 and provides
valuable insights into the properties of star clusters.
