  We consider a conformally invariant version of the Calder\'on problem, where
the objective is to determine the conformal class of a Riemannian manifold with
boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The
main result states that a locally conformally real-analytic manifold in
dimensions $\geq 3$ can be determined in this way, giving a positive answer to
an earlier conjecture by Lassas and Uhlmann (2001). The proof proceeds as in
the standard Calder\'on problem on a real-analytic Riemannian manifold, but new
features appear due to the conformal structure. In particular, we introduce a
new coordinate system that replaces harmonic coordinates when determining the
conformal class in a neighborhood of the boundary.
