  Hypervelocity stars (HVSs) unbound to the Galaxy can be formed with extreme
stellar interactions. Observational evidence comes from measurements of radial
velocities (RVs) of objects crossing the Galactic halo and of tangential
velocities based on high proper motions (HPMs) and distances of relatively
nearby stars. I searched for new HVS candidates and reviewed known objects
using their Gaia astrometric measurements. Candidates were selected with
significant Gaia parallaxes of >0.1mas, proper motions of >20mas/yr, and
computed Galactocentric tangential velocities $vtan\_g$>500km/s. The DR2 and
DR3 samples of several thousand HVS candidates were studied with respect to
their proper motions, sky distribution, number of observations, location in
crowded fields, colour-magnitude diagrams, selection effects with magnitude,
and RVs in DR3. The most extreme ($vtan\_g$>700km/s) and nearest (within 4kpc)
72 DR3 HVS candidates were investigated with respect to detected close
neighbours, flags and astrometric quality parameters of objects of similar
magnitudes in DR3. The quality checks involved HPM objects in a global
comparison and all objects in the vicinity of each target. Spurious HPMs in the
Galactic centre region led to false HVS interpretations in Gaia DR2 and are
still present in DR3, although to a lesser extent. Otherwise there is good
agreement between the HPMs of HVS candidates in DR2 and DR3. However, HVS
candidates selected from DR2 tend to have larger parallaxes hence lower
tangential velocities in DR3. Most DR3 RVs are much lower than the tangential
velocities, indicating that the DR3 HVS candidates are still affected by
underestimated parallaxes. None of the 72 extreme nearby DR3 HVS candidates,
including three D$^6$ stars, passed all quality checks. Their tangential
velocities may turn out to be smaller, but at least some of them still appear
unbound to the Galaxy. (abbreviated)
