ABSTRACT In this work, we revisit the steady-state, spherically symmetric gas accretion problem from the non-relativistic regime to the ultrarelativistic one. We first perform a detailed comparison between the Bondi and Michel models, and show how the mass accretion rate in the Michel solution approaches a constant value as the fluid temperature increases, whereas the corresponding Bondi value continually decreases, the difference between these two predicted values becoming arbitrarily large at ultrarelativistic temperatures. Additionally, we extend the Michel solution to the case of a fluid with an equation of state corresponding to a monoatomic, relativistic gas. Finally, using general relativistic hydrodynamic simulations, we study spherical accretion on to a rotating black hole, exploring the influence of the black hole spin on the mass accretion rate, the flow morphology and characteristics, and the sonic surface. The effect of the black hole spin becomes more significant as the gas temperature increases and as the adiabatic index γ stiffens. For an ideal gas in the ultrarelativistic limit (γ = 4/3), we find a reduction of 10 per cent in the mass accretion rate for a maximally rotating black hole compared to a non-rotating one, while this reduction is of up to 50 per cent for a stiff fluid (γ = 2).