We propose a new impedance imaging method, electromagnetic impedance tomography (EMIT), in which the boundary electric potential measurements in electrical impedance tomography (EIT) are augmented by measurements of the exterior magnetic field induced by the currents excited in the object by the standard EIT procedures. These magnetic measurements can be obtained reliably and inexpensively by simple pickup coils located around the imaged cross section. We derive expressions for the forward problem and for the Jacobian of the measurements, and propose an iterative reconstruction algorithm using a squared error cost function. The performance of EMIT and EIT is compared in numerical simulations using a finite-element model for the conductivity distribution of several phantoms. Evaluation of the rank and condition of the Jacobian demonstrates that the additional magnetic measurements provided by a few pickup coils in EMIT turn an underdetermined EIT problem into a well-posed one with reasonable condition, or significantly improve the conditioning of the EIT problem when it is already fully determined. Reconstructions of various phantoms reveal that EMIT provides particularly significant visual and quantitative improvement (threefold to tenfold reduction in the root-mean-squared error) in the sensitivity at the center of the object, which is the area most difficult to image using EIT.