Electrical impedance tomography and Calderón's problem

We survey mathematical developments in the inverse method of electrical impedance tomography which consists in determining the electrical properties of a medium by making voltage and current measurements at the boundary of the medium. In the mathematical literature, this is also known as Calderón's problem from Calderón's pioneer contribution (Calderón 1980 Seminar on Numerical Analysis and its Applications to Continuum Physics (Río de Janeiro, 1980) p 65 (Soc. Brasil. Mat.)). We concentrate this review around the topic of complex geometrical optics solutions that have led to many advances in the field. In the last section, we review some counterexamples to Calderón's problems that have attracted a lot of interest because of connections with cloaking and invisibility. (topical review)