We present sensitive 850$mu$m imaging of the COSMOS field using 640hr of new and archival observations taken with SCUBA-2 at the East Asian Observatorys James Clerk Maxwell Telescope. The SCUBA-2 COSMOS survey (S2COSMOS) achieves a median noise level of $sigma_{850mu{mathrm{m}}}$=1.2mJy/beam over an area of 1.6 sq. degree (MAIN; HST/ACS footprint), and $sigma_{850mu{mathrm{m}}}$=1.7mJy/beam over an additional 1 sq. degree of supplementary (SUPP) coverage. We present a catalogue of 1020 and 127 sources detected at a significance level of >4$sigma$ and >4.3$sigma$ in the MAIN and SUPP regions, respectively, corresponding to a uniform 2% false-detection rate. We construct the single-dish 850$mu$m number counts at $S_{850}$>2mJy and show that these S2COSMOS counts are in agreement with previous single-dish surveys, demonstrating that degree-scale fields are sufficient to overcome the effects of cosmic variance in the $S_{850}$=2-10mJy population. To investigate the properties of the galaxies identified by S2COSMOS sources we measure the surface density of near-infrared-selected galaxies around their positions and identify an average excess of 2.0$pm$0.2 galaxies within a 13$$ radius (~100kpc at $z$~2). The bulk of these galaxies represent near-infrared-selected SMGs and/or spatially-correlated sources and lie at a median photometric redshift of $z$=2.0$pm$0.1. Finally, we perform a stacking analysis at sub-millimeter and far-infrared wavelengths of stellar-mass-selected galaxies ($M_{star}$=10$^{10}$-10$^{12}{rm M_{odot}}$) from $z$=0-4, obtaining high-significance detections at 850um in all subsets (SNR=4-30), and investigate the relation between far-infrared luminosity, stellar mass, and the peak wavelength of the dust SED. The publication of this survey adds a new deep, uniform sub-millimeter layer to the wavelength coverage of this well-studied COSMOS field.
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