  Stellar distances constitute a foundational pillar of astrophysics. The
publication of 1.47 billion stellar parallaxes from Gaia is a major
contribution to this. Yet despite Gaia's precision, the majority of these stars
are so distant or faint that their fractional parallax uncertainties are large,
thereby precluding a simple inversion of parallax to provide a distance. Here
we take a probabilistic approach to estimating stellar distances that uses a
prior constructed from a three-dimensional model of our Galaxy. This model
includes interstellar extinction and Gaia's variable magnitude limit. We infer
two types of distance. The first, geometric, uses the parallax together with a
direction-dependent prior on distance. The second, photogeometric, additionally
uses the colour and apparent magnitude of a star, by exploiting the fact that
stars of a given colour have a restricted range of probable absolute magnitudes
(plus extinction). Tests on simulated data and external validations show that
the photogeometric estimates generally have higher accuracy and precision for
stars with poor parallaxes. We provide a catalogue of 1.47 billion geometric
and 1.35 billion photogeometric distances together with asymmetric uncertainty
measures. Our estimates are quantiles of a posterior probability distribution,
so they transform invariably and can therefore also be used directly in the
distance modulus (5log10(r)-5). The catalogue may be downloaded or queried
using ADQL at various sites (see
http://www.mpia.de/homes/calj/gedr3_distances.html) where it can also be
cross-matched with the Gaia catalogue.
