We present the catalogue and basic 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 ~1{deg}-diameter field. We combine our precise ALMA positions with the extensive multiwavelength coverage in the UDS field which yields spectral energy distributions for our SMGs and a median redshift of z_phot_=2.61+/-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 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_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 per cent 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 z~2, finding reasonable agreement - in support of an evolutionary connection between these two classes of massive galaxy.