© Springer Science+Business Media New York 2011, 2015. This chapter reviews the state of the art and the current open problems in electrical impedance tomography (EIT), which seeks to recover the conductivity (or conductivity and permittivity) of the interior of a body from knowledge of electrical stimulation and measurements on its surface. This problem is also known as the inverse conductivity problem and its mathematical formulation is due to A. P. Calderón, who wrote in 1980, the first mathematical formulation of the problem, ¿On an inverse boundary value problem.¿ EIT has interesting applications in fields such as medical imaging (to detect air and fluid flows in the heart and lungs and imaging of the breast and brain) and geophysics (detection of conductive mineral ores and the presence of ground water). It is well known that this problem is severely ill-posed, and thus this chapter is devoted to the study of the uniqueness, stability, and reconstruction of the conductivity from boundary measurements. A detailed distinction between the isotropic and anisotropic case is made, pointing out the major difficulties with the anisotropic case. The issues of global and local measurements are studied, noting that local measurements are more appropriate for practical applications such as screening for breast cancer.