A systematic, computational study of the Gilbert damping in Co|Ni, Co|Pd, and Co|Pt multilayers is carried out using first-principles scattering calculations. The damping we find shows little temperature dependence and agrees well with experimental findings only when the interface damping enhancement is taken into account. In the absence of this extrinsic enhancement, the intrinsic damping of bulklike Co|Ni multilayers increases linearly with increasing ratio of Ni to Co layers. Though spin-orbit coupling is the common origin of magnetocrystalline anisotropy and magnetization damping, the latter is not sensitive to the interface roughness that affects magnetic anisotropy strongly.