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BANYAN $\Sigma$ is a new Bayesian algorithm to identify members of youngstellar associations within 150 pc of the Sun. It includes 27 youngassociations with ages in the range ~1-800 Myr, modelled with multivariateGaussians in 6-dimensional XYZUVW space. It is the first such multi-associationclassification tool to include the nearest sub-groups of the Sco-Cen OBstar-forming region, the IC 2602, IC 2391, Pleiades and Platais 8 clusters, andthe $\rho$ Ophiuchi, Corona Australis, and Taurus star-formation regions. Amodel of field stars is built from a mixture of multivariate Gaussians based onthe Besan\c{c}on Galactic model. The algorithm can derive membershipprobabilities for objects with only sky coordinates and proper motion, but canalso include parallax and radial velocity measurements, as well asspectrophotometric distance constraints from sequences in color-magnitude orspectral type-magnitude diagrams. BANYAN $\Sigma$ benefits from an analyticalsolution to the Bayesian marginalization integrals that makes it more accurateand significantly faster than its predecessor BANYAN II. A contamination versushit rate analysis is presented and demonstrates that BANYAN $\Sigma$ achieves abetter classification performance than other moving group classification tools,especially in terms of cross-contamination between young associations. Anupdated list of bona fide members in the 27 young associations, augmented bythe Gaia-DR1 release, are presented. This new tool will make it possible toanalyze large data sets such as the upcoming Gaia-DR2 to identify new youngstars. IDL and Python versions of BANYAN $\Sigma$ are made available with thispublication. (shortened)

BANYAN. XI. The BANYAN $\Sigma$ multivariate Bayesian algorithm to identify members of young associations within 150 pc