ABSTRACT We examine the co-evolution of galaxies and supermassive black holes in the simba cosmological hydrodynamic simulation. simba grows black holes via gravitational torque-limited accretion from cold gas and Bondi accretion from hot gas, while feedback from black holes is modelled in radiative and jet modes depending on the Eddington ratio (fEdd). simba shows generally good agreement with local studies of black hole properties, such as the black hole mass–stellar velocity dispersion (MBH–σ) relation, the black hole accretion rate versus star formation rate (BHAR–SFR), and the black hole mass function. MBH–σ evolves such that galaxies at a given MBH have higher σ at higher redshift, consistent with no evolution in MBH–M⋆. For $M_{\rm BH}\lesssim 10^8\, {\rm M}_{\odot }$, fEdd is anticorrelated with MBH since the BHAR is approximately independent of MBH, while at higher masses fEdd–MBH flattens and has a larger scatter. BHAR versus SFR is invariant with redshift, but fEdd drops steadily with time at a given MBH, such that all but the most massive black holes are accreting in a radiatively efficient mode at $z\gtrsim 2$. The black hole mass function amplitude decreases with redshift and is locally dominated by quiescent galaxies for MBH > 108 M⊙, but for $z\gtrsim 1$ star-forming galaxies dominate at all MBH. The z = 0 fEdd distribution is roughly lognormal with a peak at $f_{\rm Edd}\lesssim 0.01$ as observed, shifting to higher fEdd at higher redshifts. Finally, we study the dependence of black hole properties with H i content and find that the correlation between gas content and SFR is modulated by black hole properties, such that higher SFR galaxies at a given gas content have smaller black holes with higher fEdd.