The uniaxial magnetic anisotropy energy (MAE) of L 10 FePt and Fe1-x Mnx Pt, x=0-0.25, was studied from first principles using two fully relativistic computational methods, the full-potential linear muffin-tin orbitals method and the exact muffin-tin orbitals method. It was found that the large MAE of 2.8 meV/f.u. is caused by a delicate interaction between the Fe and Pt atoms, where the large spin-orbit coupling of the Pt site and the hybridization between Fe 3d and Pt 5d states is crucial. The effect of random order on the MAE was modeled by mutual alloying of the sublattices within the coherent potential approximation (CPA), and a strong dependence of the MAE on the degree of chemical long-range order was found. The alloying of FePt with Mn was investigated with the virtual crystal approximation and the CPA as well as supercell calculations. The MAE increases up to 33% within the concentration range studied here, an effect that is attributed to band filling. Furthermore, the dependence of the MAE on the structural properties was studied. © 2005 The American Physical Society.