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Abstract. We survey some of the results obtained during the last 40 years on Calderon inverse problem: Can one determine the electrical conductivity in the interior of a medium by making voltage and current measurements at the boundary? In dimension three or higher this is equivalent to a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by making appropriate measurements of the boundary values of harmonic functions? We will mention several open problems.