Thin magnetic films deposited on single-crystal substrates are well known to present magnetic properties that cannot be found in bulk solids. Thin Co films grown on stepped Cu(11n) surface show a step-induced uniaxial magnetic anisotropy which has been discussed as a magnetoelastic effect or within the Neel model considering missing bonds at the step edge. We present a detailed study of the crystallographic structure and morphology of Co/Cu(1111) films by scanning tunneling microscopy (STM) and surface extended X-rays absorption fine structure (SEXAFS). We show that from 3 to 8 monolayers (ML), Co adopts a face centred tetragonal structure like on a flat Cu(001) substrate. For I ML, the first nearest neighbour shell is the same as in Cu in all directions, showing a step-induced stress in the [001] direction. This structure is related to large islands on the surface. Moreover. at low coverage, STM images show a restructuring of the substrate in order to reduce the surface energy. For 8 ML, we retrieve straight steps but the terraces are about two times larger than on the clean copper. This morphology may be stabilised by the tetragonal structure. These results show that the step-induced anisotropy does not originate from an in-plane structural anisotropy but is more likely due to the shape anisotropy (i.e. morphology of the Co films). (C) 2002 Elsevier Science B.V. All rights reserved.