The origin of the magnetic anisotropy of the antiferromagnetic (AF) layer and the role of it on the magnetization process of exchange-coupled ferromagnetic/AF bilayers are discussed. Through the magnetic torque analysis of a polycrystalline Ni-Fe/Mn-Ir bilayer and a pseudo-single crystalline Ni-Fe/Mn-Ni bilayer, the magnetocrystalline anisotropy of the antiferromagnet is strongly suggested to be the origin of the magnetic anisotropy of the AF layer. The single spin ensemble model is newly introduced for polycrystalline bilayers, taking into account the two-dimensional random distribution of the magnetic-anisotropy axes of AF grains. The mechanism of the reversible inducement of exchange anisotropy along desirable directions, as achieved by a field cooling procedure, is successfully elucidated with the new model. According to the simple exchange-anisotropy model established by Meiklejohn and Bean (1957), magnetic torque curves are analysed for a pseudo-single crystalline bilayer, the AF layer thickness of which is less than the critical value necessary to induce an exchange bias field. The anisotropy energy of Mn82Ni18, determined from the saturation torque amplitude, is 3.2 × 105 erg cm-3 with two-fold symmetry in (110), 1.4 × 104 erg cm-3 with four-fold symmetry in (001), and 7 × 103 erg cm-3 with six-fold symmetry in (111), respectively.