Abstract We will consider in this talk the problem arising in Electrical Impedance Tomography (EIT) of determining the conductivity of a body by making voltage and current measurements at the boundary of the body. The information is encoded in the so-called Dirichlet-to-Neumann (DN) map. Significant progress has been made in the last twenty years or so in both the theoretical and applied aspects of EIT. In particular the construction of complex geometrical optics solutions has led to many results. These results have for the most part assumed that we can make the measurements on the whole boundary. In this talk we will survey recent progress on the determination of a conductivity by making electrical measurements on part of the boundary.