  We present a catalogue of 362 million stellar parameters, distances, and
extinctions derived from Gaia's early third data release (EDR3) cross-matched
with the photometric catalogues of Pan-STARRS1, SkyMapper, 2MASS, and AllWISE.
The higher precision of the Gaia EDR3 data, combined with the broad wavelength
coverage of the additional photometric surveys and the new stellar-density
priors of the {\tt StarHorse} code allow us to substantially improve the
accuracy and precision over previous photo-astrometric stellar-parameter
estimates. At magnitude $G=14\, (17)$, our typical precisions amount to 3%
(15%) in distance, 0.13 mag (0.15 mag) in $V$-band extinction, and 140 K (180
K) in effective temperature. Our results are validated by comparisons with open
clusters, as well as with asteroseismic and spectroscopic measurements,
indicating systematic errors smaller than the nominal uncertainties for the
vast majority of objects. We also provide distance- and extinction-corrected
colour-magnitude diagrams, extinction maps, and extensive stellar density maps
that reveal detailed substructures in the Milky Way and beyond. The new density
maps now probe a much greater volume, extending to regions beyond the Galactic
bar and to Local Group galaxies, with a larger total number density. We publish
our results through an ADQL query interface ({\tt gaia.aip.de}) as well as via
tables containing approximations of the full posterior distributions. Our
multi-wavelength approach and the deep magnitude limit make our results useful
also beyond the next Gaia release, DR3.
