We present a spectral atlas of the post-main-sequence population of the most massive Galactic globular cluster, omega Centauri. Spectra were obtained of more than 1500 stars selected as uniformly as possible from across the (B, B-V) colour-magnitude diagram of the proper motion cluster member candidates of van Leeuwen et al. (2000, Cat. <J/A+A/360/472>). The spectra were obtained with the 2dF multifibre spectrograph at the Anglo-Australian Telescope, and cover the approximate range {lambda}~3840-3940{AA} at a resolving power of {lambda}/{Delta}{lambda}~=2000. This constitutes the most comprehensible spectroscopic survey of a globular cluster. We measure the radial velocities, effective temperatures, metallicities and surface gravities by fitting ATLAS9 stellar atmosphere models. We analyse the cluster membership and stellar kinematics, interstellar absorption in the Ca II K line at 3933{AA}, the RR Lyrae instability strip and the extreme horizontal branch, the metallicity spread and bimodal CN abundance distribution of red giants, nitrogen and s-process enrichment, carbon stars, pulsation-induced Balmer line emission on the asymptotic giant branch (AGB), and the nature of the post-AGB and UV-bright stars. Membership is confirmed for the vast majority of stars, and the radial velocities clearly show the rotation of the cluster core. We identify long-period RR Lyrae-type variables with low gravity, and low-amplitude variables coinciding with warm RR Lyrae stars. A barium enhancement in the coolest red giants indicates that third dredge-up operates in AGB stars in omega Cen. This is distinguished from the pre-enrichment by more massive AGB stars, which is also seen in our data. The properties of the AGB, post-AGB and UV-bright stars suggest that red giant branch (RGB) mass loss may be less efficient at very low metallicity, [Fe/H]<<1, increasing the importance of mass loss on the AGB. The catalogue and spectra are made available via Centre Donnees de Strasbourg (CDS).