We present the results of the HIGHz Arecibo survey, which measured the HI content of 39 galaxies at redshift z > 0.16 selected from the Sloan Digital Sky Survey. These are all actively star-forming, disc-dominated systems in relatively isolated environments, with stellar and HI masses larger than 1010 M⊙ and redshifts 0.17 ≤ z ≤ 0.25. Our sample includes not only the highest-redshift detections of HI emission from individual galaxies to date, but also some of the most HI-massive systems known. Despite being exceptionally large, the HI reservoirs of these galaxies are consistent with what is expected from their ultraviolet and optical properties. This, and the fact that the galaxies lie on the baryonic Tully-Fisher relation, suggests that HIGHz systems are rare, scaled-up versions of local disc galaxies.We show that the most HI-massive galaxies discovered in the Arecibo Legacy Fast ALFA survey are the local analogues of HIGHz, and discuss the possible connection between our sample and the turbulent, gas-rich discs identified at z ~ 1. The HIGHz sample provides a first glimpse into the properties of the massive, HI-rich galaxies that will be detected at higher redshifts by the next generation HI surveys with the Square Kilometre Array and its pathfinders. © 2014 The Authors.