In its geometric formulation, the anisotropic Calderón inverse problem consists in showing that the metric of a compact Riemannian manifold with boundary is uniquely determined (up to some natural gauge equivalences) from the knowledge of the Dirichlet-to-Neumann map for the Laplacian, that is the map that assigns to data prescribed on the boundary the normal derivative of the corresponding solution to Laplace's equation. While the Calderón inverse problem is still open in its full generality, there are a number of results providing either an affirmative answer or counterexamples, depending on which special assumptions are made about the background geometry. After introducing the Calderón inverse problem, I will review some of these uniqueness and non-uniqueness results.