The crystallographic structure and magnetic anisotropy have been investigated in Co/Pt multilayers deposited by direct current magnetron sputtering on two different buffer layers, Ta and Pt. Detail theoretical and experimental investigations reveal the presence of three effects: magneto-elastic,interface and shape anisotropies, and their competition results in three distinct regions based on the Co layer thickness (tCo). In the region I, with tCo < 6 A° , the coherent lattice strain modifies the volume anisotropy through magneto-elastic contribution and it leads to overall positive volume anisotropy energy. With further increase in tCo, lattice mismatch initiates and this results in a decrease in magneto-elastic effect and it enhances the interface anisotropy. The presence of both diminished magneto-elastic effect and enhanced interface anisotropy results in an intermediate region (region II) with negative volume anisotropy energy which is relatively wider (6 ˚A < tCo < 12 A˚ ) for multilayer stacks with Ta underlayer. In the region III, with tCo > 12 A˚ , the magneto-elastic effect emerges as an interface phenomena and the shape anisotropy becomes dominant.