  We compare the Gaia DR2 and Gaia EDR3 performances in the study of the
Magellanic Clouds and show the clear improvements in precision and accuracy in
the new release. We also show that the systematics still present in the data
make the determination of the 3D geometry of the LMC a difficult endeavour;
this is at the very limit of the usefulness of the Gaia EDR3 astrometry, but it
may become feasible with the use of additional external data.
  We derive radial and tangential velocity maps and global profiles for the LMC
for the several subsamples we defined. To our knowledge, this is the first time
that the two planar components of the ordered and random motions are derived
for multiple stellar evolutionary phases in a galactic disc outside the Milky
Way, showing the differences between younger and older phases. We also analyse
the spatial structure and motions in the central region, the bar, and the disc,
providing new insights into features and kinematics.
  Finally, we show that the Gaia EDR3 data allows clearly resolving the
Magellanic Bridge, and we trace the density and velocity flow of the stars from
the SMC towards the LMC not only globally, but also separately for young and
evolved populations. This allows us to confirm an evolved population in the
Bridge that is slightly shift from the younger population. Additionally, we
were able to study the outskirts of both Magellanic Clouds, in which we
detected some well-known features and indications of new ones.
