We present the catalogue and properties of sources in AS2UDS, an 870-$mu$m continuum survey with the Atacama Large Millimetre/sub-millimetre Array (ALMA) of 716 single-dish sub-millimetre sources detected in the UKIDSS/UDS field by the SCUBA-2 Cosmology Legacy Survey. In our sensitive ALMA follow-up observations we detect 708 sub-millimetre galaxies (SMGs) at $>$,4.3$sigma$ significance across the $sim$,1-degree diameter field. We combine our precise ALMA positions with the extensive multi-wavelength coverage in the UDS field to fit the spectral energy distributions of our SMGs to derive a median redshift of $z_{rm phot}=$,2.61$pm$0.09. This large sample reveals a statistically significant trend of increasing sub-millimetre flux with redshift suggestive of galaxy downsizing. 101 ALMA maps do not show a $>$,4.3$sigma$ SMG, but we demonstrate from stacking {it Herschel} SPIRE observations at these positions, that the vast majority of these blank maps correspond to real single-dish sub-millimetre sources. We further show that these blank maps contain an excess of galaxies at $z_{rm phot}=$,1.5--4 compared to random fields, similar to the redshift range of the ALMA-detected SMGs. In addition, we combine X-ray and mid-infrared active galaxy nuclei activity (AGN) indicators to yield a likely range for the AGN fraction of 8--28,% in our sample. Finally, we compare the redshifts of this population of high-redshift, strongly star-forming galaxies with the inferred formation redshifts of massive, passive galaxies being found out to $zsim$,2, finding reasonable agreement -- in support of an evolutionary connection between these two classes of massive galaxy.
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