The performance of some GaAs devices is impaired by a high surface recombination rate brought about by surface states which pin the Fermi level in the middle of the band gap. Passivation of the surface with S or Se can significantly reduce this problem. The pinning on the clean GaAs(001)-(2X4) surface of molecular beam epitaxy grown material is understood from previous scanning tunneling microscopy (STM) studies. We have now used the STM to study the reaction of Se with the GaAs surface and tunneling spectroscopy to determine the change in the Fermi-level position as a result of the Se deposition. Under appropriate deposition and annealing conditions Se forms a well ordered (2X1) reconstructed surface on GaAs(001). The Fermi level on this surface is found to be within 150 meV of the conduction band minimum on heavily n-doped (approximately 6X10(18) cm-3 Si) material. STM images show that the kinks in the dimer-vacancy rows of the GaAs(001)-(2X4) reconstruction, which pin the Fermi level on n-type material, are removed when Se is deposited and so the Fermi-level pinning is greatly reduced. In addition, the density of defects (surface vacancies, steps, etc.) is relatively low which will reduce pinning on both n- and p-type material. However, the GaAs(001):Se-(2X1) surface is not sufficiently chemically inert to prevent some oxidation (and presumably pinning) on exposure to the atmosphere. We propose a structural model for the GaAs(001):Se-(2X1) surface that is based on our STM determination of the surface layer composition, published photoemission data which show Se to go below the surface and displace arsenic, and that satisfies electron counting.