Structure and magnetic properties of sputter-prepared Co(5 nm)/MnPt bilayer films with Ta underlayer and various thicknesses of MnPt layer (t(MnPt)) have been studied. Ta underlayer is effective in promoting the ordering of MnPt layer, flattening the surface and interface, and therefore leading to the improvement of the exchange bias field (H-E). Nevertheless, higher annealing temperature leads to the interdiffusion and decrease H-E. Besides, low H-E of 8 kA/m with higher coercivity for the sample with thinner t(MnPt) = 10 nm might be related lower magnetic anisotropy energy for MnPt layer due to too thin antiferromagnetic layer, and therefore, H-E is increased to 13.8-20.6 kA/m with increasing t(MnPt) to 30-100 nm. Larger grain size distribution and roughened interface found for the sample with thicker t(MnPt) = 100 nm may result in reduced H-E. This study provides useful information to fabricate exchange-bias Co/L1(0)-MnPt system.