NGC 3532 is an extremely rich open cluster embedded in the Galactic disc, hitherto lacking a comprehensive, documented membership list. We provide membership probabilities from new radial velocity observations of solar-type and low-mass stars in NGC 3532, in part as a prelude to a subsequent study of stellar rotation in the cluster. Using extant optical and infra-red photometry we constructed a preliminary photometric membership catalogue, consisting of 2230 dwarf and turn-off stars. We selected 1060 of these for observation with the AAOmega spectrograph at the 3.9m Anglo-Australian Telescope and 391 stars for observations with the Hydra-South spectrograph at the 4m Victor Blanco Telescope, obtaining spectroscopic observations over a decade for 145 stars. We measured radial velocities for our targets through cross-correlation with model spectra and standard stars, and supplemented them with radial velocities for 433 additional stars from the literature. We also measured logg, Teff, and [Fe/H] from the AAOmega spectra. The radial velocity distribution emerging from the observations is centred at 5.43+/-0.04km/s and has a width (standard deviation) of 1.46km/s. Together with proper motions from Gaia DR2 we find 660 exclusive members, of which five are likely binary members. The members are distributed across the whole cluster sequence, from giant stars to M dwarfs, making NGC~3532 one of the richest Galactic open clusters known to date, on par with the Pleiades. From further spectroscopic analysis of 153 dwarf members we find the metallicity to be marginally sub-solar, with [Fe/H]=-0.07+/-0.10. We confirm the extremely low reddening of the cluster, E_B-V_=0.034+/-0.012mag, despite its location near the Galactic plane. Exploiting trigonometric parallax measurements from Gaia DR2 we find a distance of 484^+35^_-30_pc [(m-M)_0_=8.42+/-0.14mag]. Based on the membership we provide an empirical cluster sequence in multiple photometric passbands. A comparison of the photometry of the measured cluster members with several recent model isochrones enables us to confirm the 300Myr cluster age. However, all of the models evince departures from the cluster sequence in particular regions, especially in the lower mass range.