We will consider in these introductory lectures the inverse boundary problem of Electrical Impedance Tomography (EIT). This inverse method consists in determining the electrical conductivity inside a body by making voltage and current measurements at the boundary. The boundary information is encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is to determine the coefficients of the conductivity equation (an elliptic partial differential equation) knowing the DN map. We will also consider the anisotropic case which can be formulated, in dimension three or larger, as the question of determining a Riemannian metric from the associated DN map. We will discuss a connection of this latter problem with the boundary rigidity problem which will be the topic of C. Croke's lectures. In this case the information is encoded in the boundary distance function which measures the lengths of geodesics joining points in the boundary of a compact Riemannian manifold with boundary.