  BANYAN $\Sigma$ is a new Bayesian algorithm to identify members of young
stellar associations within 150 pc of the Sun. It includes 27 young
associations with ages in the range ~1-800 Myr, modelled with multivariate
Gaussians in 6-dimensional XYZUVW space. It is the first such multi-association
classification tool to include the nearest sub-groups of the Sco-Cen OB
star-forming region, the IC 2602, IC 2391, Pleiades and Platais 8 clusters, and
the $\rho$ Ophiuchi, Corona Australis, and Taurus star-formation regions. A
model of field stars is built from a mixture of multivariate Gaussians based on
the Besan\c{c}on Galactic model. The algorithm can derive membership
probabilities for objects with only sky coordinates and proper motion, but can
also include parallax and radial velocity measurements, as well as
spectrophotometric distance constraints from sequences in color-magnitude or
spectral type-magnitude diagrams. BANYAN $\Sigma$ benefits from an analytical
solution to the Bayesian marginalization integrals that makes it more accurate
and significantly faster than its predecessor BANYAN II. A contamination versus
hit rate analysis is presented and demonstrates that BANYAN $\Sigma$ achieves a
better classification performance than other moving group classification tools,
especially in terms of cross-contamination between young associations. An
updated list of bona fide members in the 27 young associations, augmented by
the Gaia-DR1 release, are presented. This new tool will make it possible to
analyze large data sets such as the upcoming Gaia-DR2 to identify new young
stars. IDL and Python versions of BANYAN $\Sigma$ are made available with this
publication. (shortened)
