  The goal of this paper is to demonstrate the outstanding quality of the
second data release of the Gaia mission and its power for constraining many
different aspects of the dynamics of the satellites of the Milky Way. We focus
here on determining the proper motions of 75 Galactic globular clusters, nine
dwarf spheroidal galaxies, one ultra-faint system, and the Large and Small
Magellanic Clouds. Using data extracted from the Gaia archive, we derived the
proper motions and parallaxes for these systems, as well as their
uncertainties. We demonstrate that the errors, statistical and systematic, are
relatively well understood. We integrated the orbits of these objects in three
different Galactic potentials, and characterised their properties. We present
the derived proper motions, space velocities, and characteristic orbital
parameters in various tables to facilitate their use by the astronomical
community. Our limited and straightforward analyses have allowed us for example
to (i) determine absolute and very precise proper motions for globular
clusters; (ii) detect clear rotation signatures in the proper motions of at
least five globular clusters; (iii) show that the satellites of the Milky Way
are all on high-inclination orbits, but that they do not share a single plane
of motion; (iv) derive a lower limit for the mass of the Milky Way of
9.8^{+6.7}_{-2.7} x 10^{11} Msun based on the assumption that the Leo I dwarf
spheroidal is bound; (v) derive a rotation curve for the Large Magellanic Cloud
based solely on proper motions that is competitive with line-of-sight velocity
curves, now using many orders of magnitude more sources; and (vi) unveil the
dynamical effect of the bar on the motions of stars in the Large Magellanic
Cloud. All these results highlight the incredible power of the Gaia astrometric
mission, and in particular of its second data release.
