The growth of Bi on both the In-terminated (A) face and the As-terminated (B) face of InAs(111) has been investigated by low-energy electron diffraction, scanning tunnelling microscopy, and photoelectron spectroscopy using synchrotron radiation. The changes upon Bi deposition of the In 4d and Bi 5d(5/2) photoelectron signals allow to get a comprehensive picture of the Bi/InAs(111) interface. From the early stage the Bi growth on the A face is epitaxial, contrary to that on the B face that proceeds via the formation of islands. Angle-resolved photoelectron spectra show that the electronic structure of a Bi deposit of approximate to 10 bi-layers on the A face is identical to that of bulk Bi, while more than approximate to 30 bi-layers are needed for the B face. Both bulk and surface electronic states observed are well accounted for by fully relativistic ab initio calculations performed using the one-step model of photoemission. These calculations are used to analyse the dichroic photoemission data recorded in the vicinity of the Fermi level around the (Gamma) over bar point of the Brillouin zone.