Abstract.We study integral expressions of electromagnetic multipole moments of arbitrary order in Cartesian coordinates. The volume and surface integrals of charge-induced and current-induced multipole moment tensors are formulated and the relationship between them is discussed. Full surface integral expressions for the multipole moment are also obtained. We further extend the formulation to introduce another kind of dipole moment, which is similar to the charge-induced and current-induced multipole moments and is found in a vector decomposition formula. † This is an Author's Original Manuscript of an article whose final and definitive form, the Version of Record, has been published in the