Electrical impedance tomography systems apply currents to the surface of a body and measure the resulting voltages. From these electrical measurements approximate reconstructions of the electrical conductivity, permittivity , and magnetic permeability are made of the interior of the body. These interior electrical properties are displayed and can be used for monitoring heart and lung function as well as for improving the diagnosis of breast cancer . It will be shown how problems in the design of electrical impedance tomography systems can be formulated as mathematical problems about the Neumann to Dirichlet map for Maxwell's equations. It will be shown how the reconstruction of interior electrical properties gives rise to inverse boundary value problems for Maxwell's equations. Images and movies of heart and lung function as well as impedance spectra of breast cancers that were made by RPI's ACT systems will be shown. If time permits it will be explained how Nachman's d-bar method of inverse scattering has been used to image heart and lung function.