ABSTRACT We use the state-of-the-art semi-analytic galaxy formation model, shark, to investigate the physical processes involved in dictating the shape, scatter, and evolution of the Hi–halo mass (HIHM) relation at 0 ≤ z ≤ 2. We compare shark with Hi clustering and spectral stacking of the HIHM relation derived from observations finding excellent agreement with the former and a deficiency of Hi in shark at Mvir ≈ 1012–13 M⊙ in the latter. In shark, we find that the Hi mass increases with the halo mass up to a critical mass of ≈1011.8 M⊙; between ≈1011 and 1013 M⊙, the scatter in the relation increases by 0.7 dex and the Hi mass decreases with the halo mass on average (till $M_{\rm vir}∼ 10^{12.5}\, \rm M_{⊙ }$, after which it starts increasing); at $M_{\rm vir}\gtrsim 10^{13}\, \rm M_{⊙ }$, the Hi content continues to increase with increasing halo mass, as a result of the increasing Hi contribution from satellite galaxies. We find that the critical halo mass of ≈1012 M⊙ is set by feedback from active galactic nuclei (AGNs) which affects both the shape and scatter of the HIHM relation, with other physical processes playing a less significant role. We also determine the main secondary parameters responsible for the scatter of the HIHM relation, namely the halo spin parameter at ${M}_{\rm vir}\, \lt $ 1011.8 M⊙, and the fractional contribution from substructure to the total halo mass ($M_{\rm h}^{\rm sat}/M_{\rm vir}$) for ${M}_{\rm vir}\, \gt $ 1013 M⊙. The scatter at 1011.8 M⊙$\lt \, {M}_{\rm vir}\, \lt $ 1013 M⊙ is best described by the black hole-to-stellar mass ratio of the central galaxy, reflecting the relevance of AGN feedback. We present a numerical model to populate dark matter-only simulations with Hi at 0 ≤ z ≤ 2 based solely on halo parameters that are measurable in such simulations.