The electronic and magnetic structures of Mn films with thicknesses of up to 12 monolayers on Fe(001) are investigated by means of first-principles calculations, with a focus on the thicker films. The Mn films show an in-layer antiferromagnetic structure and couple antiferromagnetically to the Fe substrate. The magnetic moments at the Mn surface and at the Mn-Fe interface are significantly enhanced, as compared to the respective bulk values. Several surface states are found at the centre of the two-dimensional Brillouin zone, some of which might explain recent experimental scanning tunneling spectroscopy results.