  Gaia Data Release 2 (Gaia DR2) contains results for 1693 million sources in
the magnitude range 3 to 21 based on observations collected by the European
Space Agency Gaia satellite during the first 22 months of its operational
phase. We describe the input data, models, and processing used for the
astrometric content of Gaia DR2, and the validation of these results performed
within the astrometry task. Some 320 billion centroid positions from the
pre-processed astrometric CCD observations were used to estimate the five
astrometric parameters (positions, parallaxes, and proper motions) for 1332
million sources, and approximate positions at the reference epoch J2015.5 for
an additional 361 million mostly faint sources. Special validation solutions
were used to characterise the random and systematic errors in parallax and
proper motion. For the sources with five-parameter astrometric solutions, the
median uncertainty in parallax and position at the reference epoch J2015.5 is
about 0.04 mas for bright (G<14 mag) sources, 0.1 mas at G=17 mag, and 0.7 mas
at G=20 mag. In the proper motion components the corresponding uncertainties
are 0.05, 0.2, and 1.2 mas/yr, respectively. The optical reference frame
defined by Gaia DR2 is aligned with ICRS and is non-rotating with respect to
the quasars to within 0.15 mas/yr. From the quasars and validation solutions we
estimate that systematics in the parallaxes depending on position, magnitude,
and colour are generally below 0.1 mas, but the parallaxes are on the whole too
small by about 0.03 mas. Significant spatial correlations of up to 0.04 mas in
parallax and 0.07 mas/yr in proper motion are seen on small (<1 deg) and
intermediate (20 deg) angular scales. Important statistics and information for
the users of the Gaia DR2 astrometry are given in the appendices.
