  After giving a general introduction to the main known results on the
anisotropic Calder{\'o}n problem on n-dimensional compact Riemannian manifolds
with boundary, we give a motivated review of some recent non-uniqueness results
obtained in [5, 6] for the anisotropic Calder{\'o}n problem at fixed frequency,
in dimension n $\ge$ 3, when the Dirichlet and Neumann data are measured on
disjoint subsets of the boundary. These non-uniqueness results are of the
following nature: given a smooth compact connected Riemannian manifold with
boundary (M, g) of dimension n $\ge$ 3, we first show that there exist in the
conformal class of g an infinite number of Riemannian metrics gmetrics metrics
g such that their corresponding Dirichlet-to-Neumann maps at a fixed frequency
coincide when the Dirichlet data $\Gamma$D and Neumann data $\Gamma$N are
measured on disjoint sets and satisfy $\Gamma$D $\cup$ $\Gamma$N = $\partial$M.
The corresponding conformal factors satisfy a nonlinear elliptic PDE of Yamabe
type on (M, g) and arise from a natural but subtle gauge invariance of the
Calder{\'o}n when the data are given on disjoint sets. We then present
counterexamples to uniqueness in dimension n $\ge$ 3 to the anisotropic
Calder{\'o}n problem at fixed frequency with data on disjoint sets, which do
not arise from this gauge invariance. They are given by cylindrical Riemannian
manifolds with boundary having two ends, equipped with a suitably chosen warped
product metric. This survey concludes with some remarks on the case of
manifolds with corners.
