In Pt/Co multilayers ML exchange coupled to IrMn, the magnitudes of the exchange bias field H E and coercivity H C measured along the perpendicular to film direction, can be tailored by i varying the thickness of the Co layers t Co inside the ML and/or ii inserting a Pt spacer between the ML and the antiferromagnetic AFM layer. An unusual peak in the ferromagnetic FM thickness dependence of exchange bias properties is observed. This is ascribed to a reduction of the perpendicular effective magnetic anisotropy for either very small or too large values of t Co. Moreover, for low values of t Co , the insertion of an ultrathin Pt spacer between the Pt/Co ML and the IrMn brings about a significant increase of H E and H C. However, such an effect is not observed for thicker Co layers. This behavior is explained by the twofold role of the Pt spacer, i.e., it strengthens the perpendicular orientation of the Co magnetization in the ML but it also tends to reduce exchange bias due to the short-range character of the FM-AFM interactions.