BANYAN {Sigma} is a new Bayesian algorithm to identify members of young stellar associations within 150pc of the Sun. It includes 27 young associations with ages in the range ~1-800Myr, modeled with multivariate Gaussians in six-dimensional (6D) 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 IC2602, IC2391, 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 Besancon 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 over unknown radial velocities and distances 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 tools available in the literature, 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, as well as all parameters for the 6D multivariate Gaussian models for each association and the Galactic field neighborhood within 300 pc are presented.