The puzzling electronic and transport properties of the Ga(AsBi) alloy are investigated for a wide range of Bi-concentrations (x = 0–10.6%) by means of various experimental techniques in high magnetic fields (B up to 30 T): magneto-photoluminescence spectroscopy, magneto-far-infrared (FIR) absorption spectroscopy, and Hall effect measurements. Our experimental findings suggest that the strength of hybridization of the continuum states of the valence and conduction bands with the Bi-related electronic levels depends on the Bi-concentration, thus leading to band edges with a localized (for x < 6%) or band-like character (for x > 8%). We report an unusual compositional-dependence of the exciton reduced mass (µexc), whose value can be larger (for x < 6%) or smaller (for x > 8%) than in GaAs depending on the Bi-concentration. Correspondingly, the free-hole mobility (µh) decreases with increasing Bi-concentration and eventually tends to increase for x > ∼8%. The incorporation of Bi in GaAs also induces the formation of acceptor levels, which we reveal by FIR absorption spectroscopy and Hall effect measurements. The Bi-induced acceptors are characterized by an exceedingly high value of the effective ground-state g-factor (geff ∼ 15) and are responsible for the increasing p-type conductivity observed in nominally undoped Ga(AsBi) alloys with increasing Bi-concentration.