  We revisit the observational constraints on spatial curvature following
recent claims that the Planck data favour a closed Universe. We use a new and
statistically powerful Planck likelihood to show that the Planck temperature
and polarization spectra are consistent with a spatially flat Universe, though
because of a geometrical degeneracy cosmic microwave background spectra on
their own do not lead to tight constraints on the curvature density parameter
Omega_K. When combined with other astrophysical data, particularly geometrical
measurements of baryon acoustic oscillations, the Universe is constrained to be
spatially flat to extremely high precision, with Omega_ K = 0.0004 +/-0.0018 in
agreement with the 2018 results of the Planck team. In the context of
inflationary cosmology, the observations offer strong support for models of
inflation with a large number of e-foldings and disfavour models of incomplete
inflation.
