The second data release from the Gaia mission (DR2) provides a comprehensive and unprecedented picture of the motions of astronomical sources in the plane of the sky, extending from the solar neighborhood to the outer reaches of the Milky Way. I present proper-motion measurements based on Gaia DR2 for 17 ultra-faint dwarf galaxies within 100kpc of the Milky Way. I compile the spectroscopically confirmed member stars in each dwarf bright enough for Gaia astrometry from the literature, producing member samples ranging from two stars in Triangulum II to 68 stars in Bootes I. From the spectroscopic member catalogs, I estimate the proper motion of each system. I find good agreement with the proper motions derived by the Gaia collaboration for Bootes I and Leo I. The tangential velocities for 14 of the 17 dwarfs are determined to better than 50km/s, more than doubling the sample of such measurements for Milky Way satellite galaxies. The orbital pericenters are well constrained, with a mean value of 38kpc. Only one satellite, Tucana III, is on an orbit passing within 15kpc of the Galactic center, suggesting that the remaining ultra-faint dwarfs are unlikely to have experienced severe tidal stripping. As a group, the ultra-faint dwarfs are on high-velocity, eccentric, retrograde trajectories, with nearly all of them having space motions exceeding 370km/s. A large majority of the objects are currently close to the pericenters of their orbits. In a low-mass (M_vir_=0.9x10^12^M_{sun}_) Milky Way potential, eight out of the 17 galaxies lack well-defined apocenters and appear likely to be on their first infall, indicating that the Milky Way mass may be larger than previously estimated or that many of the ultra-faint dwarfs are associated with the Magellanic Clouds. The median eccentricity of the ultra-faint dwarf orbits is 0.79, similar to the values seen in numerical simulations but distinct from the rounder orbits of the more luminous dwarf spheroidals.