We present sensitive 850{mu}m imaging of the Cosmological Evolution Survey (COSMOS) field using 640hr of new and archival observations taken with SCUBA-2 at the East Asian Observatory's James Clerk Maxwell Telescope. The SCUBA-2 COSMOS survey (S2COSMOS) achieves a median noise level of {sigma}_850{mu}m_=1.2mJy/beam over an area of 1.6deg^2^ (MAIN; Hubble Space Telescope/Advanced Camera for Surveys footprint), and {sigma}_850{mu}m_=1.7mJy/beam over an additional 1deg^2^ of supplementary (supp) coverage. We present a catalog 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+/-0.2 galaxies within a 13" radius (~100kpc at z~2). The bulk of these galaxies represent near-infrared-selected submillimeter galaxies and/or spatially correlated sources and lie at a median photometric redshift of z=2.0+/-0.1. Finally, we perform a stacking analysis at submillimeter and far-infrared wavelengths of stellar-mass-selected galaxies (M_*_=10^10^-10^12^M_{sun}_) from z=0-4, obtaining high-significance detections at 850{mu}m in all subsets (signal-to-noise ratio, S/N=4-30), and investigate the relation between far-infrared luminosity, stellar mass, and the peak wavelength of the dust spectral energy distribution. The publication of this survey adds a new deep, uniform submillimeter layer to the wavelength coverage of this well-studied COSMOS field.