  Context. Although the Gaia catalogue on its own is a very powerful tool, it
is the combination of this high-accuracy archive with other archives that will
truly open up amazing possibilities for astronomical research. The advanced
interoperation of archives is based on cross-matching, leaving the user with
the feeling of working with one single data archive. The data retrieval should
work not only across data archives but also across wavelength domains. The
first step for a seamless access to the data is the computation of the
cross-match between Gaia and external surveys.
  Aims. We describe the adopted algorithms and results of the pre-computed
cross-match of the Gaia Data Release 2 (DR2) catalogue with dense surveys
(Pan-STARRS1 DR1, 2MASS, SDSS DR9, GSC 2.3, URAT-1, allWISE, PPMXL, and APASS
DR9) and sparse catalogues (Hipparcos2, Tycho-2, and RAVE 5).
  Methods. A new algorithm is developed specifically for sparse catalogues.
Improvements and changes with respect to the algorithm adopted for DR1 are
described in detail.
  Results. The outputs of the cross-match are part of the official Gaia DR2
catalogue. The global analysis of the cross-match results is also presented.
