We analyse the physical properties of a large, homogeneously selected sample of ALMA-located sub-mm galaxies (SMGs). This survey, AS2UDS, identified 707 SMGs across the ~1 sq.deg. field, including ~17 per cent, which are undetected at $K$>~25.7 mag. We interpret their ultraviolet-to-radio data using MAGPHYS and determine a median redshift of z=2.61+-0.08 (1$sigma$ range of z=1.8-3.4) with just ~6 per cent at z>4. Our survey provides a sample of massive dusty galaxies at z>~1, with median dust and stellar masses of $M_d$=(6.8+-0.3)x10$^{8}$M$_odot$ (thus, gas masses of ~10$^{11}$M$_odot$) and $M_ast=$(1.26+-0.05)x10$^{11}$M$_odot$. We find no evolution in dust temperature at a constant far-infrared luminosity across z~1.5-4. The gas mass function of our sample increases to z~2-3 and then declines at z>3. The space density and masses of SMGs suggest that almost all galaxies with $M_ast$>~3x10$^{11}$M$_odot$ have passed through an SMG-like phase. The redshift distribution is well fit by a model combining evolution of the gas fraction in halos with the growth of halo mass past a threshold of $M_h$~6x10$^{12}$M$_odot$, thus SMGs may represent the highly efficient collapse of gas-rich massive halos. We show that SMGs are broadly consistent with simple homologous systems in the far-infrared, consistent with a centrally illuminated starburst. Our study provides strong support for an evolutionary link between the active, gas-rich SMG population at z>1 and the formation of massive, bulge-dominated galaxies across the history of the Universe.
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