We show that the Dirichlet-to-Neumann operator
of the Laplacian on an open subset of the boundary of a connected
compact Einstein manifold with boundary determines the manifold up to isometries. Similarly, for connected conformally compact
Einstein manifolds of even dimension $n+1,$ we prove that the scattering matrix at energy $n$
on an open subset of its boundary determines the manifold up to isometries.