ABSTRACT We use the IllustrisTNG (TNG) simulations to explore the galaxy–halo connection as inferred from state-of-the-art cosmological, magnetohydrodynamical simulations. With the high-mass resolution and large volume achieved by combining the 100 Mpc (TNG100) and 300 Mpc (TNG300) volumes, we establish the mean occupancy of central and satellite galaxies and their dependence on the properties of the dark matter haloes hosting them. We derive best-fitting HOD parameters from TNG100 and TNG300 for target galaxy number densities of $\bar{n}_g = 0.032\,$ and $\bar{n}_g = 0.016\, h^3$ Mpc−3, respectively, corresponding to a minimum galaxy stellar mass of $M_⋆ ∼ 1.9\times 10^9\, $ and $M_⋆ ∼ 3.5\times 10^9\, {\rm M}_⊙$, respectively, in hosts more massive than $10^{11}\, {\rm M}_⊙$. Consistent with previous work, we find that haloes located in dense environments, with low concentrations, later formation times, and high angular momenta are richest in their satellite population. At low mass, highly concentrated haloes and those located in overdense regions are more likely to contain a central galaxy. The degree of environmental dependence is sensitive to the definition adopted for the physical boundary of the host halo. We examine the extent to which correlations between galaxy occupancy and halo properties are independent and demonstrate that HODs predicted by halo mass and present-day concentration capture the qualitative dependence on the remaining halo properties. At fixed halo mass, concentration is a strong predictor of the stellar mass of the central galaxy, which may play a defining role in the fate of the satellite population. The radial distribution of satellite galaxies, which exhibits a universal form across a wide range of host halo mass, is described accurately by the best-fitting NFW density profile of their host haloes.