We present ab-initio calculations of the spin-structures and the magnetic anisotropies of face-centred-cubic iron films on Cu(100) substrates with the thickness varying between one and seven monolayers. The calculations are based on standard linear-muffin-tin orbital (LMTO) calculations, as well as a tight-binding LMTO method with variable local spin quantization axes. Our calculations show that for films with more than three monolayers, the magnetic polarization of the films acquires an antiferromagnetic component, for films with more than four monolayers metastable magnetic configurations appear. For the fcc Fe monolayers we predict a perpendicular anisotropy on Cu(100) and an in-plane anisotropy on Cu(111). For the thicker layers on Cu(100) we find a non-monotonic decrease in the magnetic anisotropy energy from Delta E=-1.87 meV for the monolayer to Delta E=-0.33 meV for the seven-layer film. Perpendicular anisotropy subsists for up to seven monolayers, we expect a transition to in-plane anisotropy for 8 to 10 monolayers.