High-velocity clouds (HVCs), fast-moving ionized and neutral gas clouds found at high galactic latitudes, may play an important role in the evolution of the Milky Way. The extent of this role depends sensitively on their distances and total sky covering factor. We search for HVC absorption in Hubble Space Telescope high-resolution ultraviolet (UV) spectra of a carefully selected sample of 133 active galactic nuclei (AGN) using a range of atomic species in different ionization stages (e.g. O i, C ii, C iv, Si ii, Si iii, Si iv). This allows us to identify neutral, weakly ionized or highly ionized HVCs over several decades in H i column densities. The sky covering factor of UV-selected HVCs with |vLSR| ≥ 90 km s−1 is about 68 per cent for the Galactic sky at . About 74 per cent of the HVC directions have cm−2 and 46 per cent have cm−2. We show that our survey is essentially complete, i.e. an undetected population of HVCs with extremely low H (H i + H ii) column density is unlikely to be important for the HVC mass budget. We confirm that the predominantly ionized HVCs contain at least as much mass as the traditional H i HVCs and show that large H i HVC complexes have generally ionized envelopes extending far from the H i contours. There are also large regions of the Galactic sky that are covered with ionized high-velocity gas with little H i emission nearby. We show that the covering factors of HVCs with km s−1 drawn from the AGN and stellar (with stars at d > 3 kpc) samples are similar. This confirms that most of the HVCs are within 5–15 kpc of the sun. The HVCs with |vLSR |≳170 km s−1 are largely associated with the Magellanic Stream at b < −20° and its leading arm at b > 20° as well as other large known H i complexes. We conclude that there is no evidence in the Local Group that any galaxy shows a population of HVCs extending much farther away than 50 kpc from its host, except possibly for those tracing remnants of galaxy interaction.