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.
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