Abstract The electromagnetic form factors of the deuteron are calculated in the Skyrme approach of forming the baryons as topological solitons of meson fields. The form factors are obtained as matrix elements of the anomalous isoscalar current operator, with the chiral angle determined from the isoscalar electric form factor of the nucleon. The current operators for the deuteron are represented by the product ansatz. The magnetic form factor is dominated by the exchange-current contribution and can, depending somewhat on the isoscalar form factor used as input, be brought into agreement with the empirical value over the entire range where data are available ( q 2 ⩽ 70 fm −2 ). There are also large exchange-current contributions to the charge and quadrupole form factors. The predicted tensor polarization is similar to that obtained with conventional meson-exchange current mechanisms.