  We consider Calderon's inverse problem with partial data in dimensions $n
\geq 3$. If the inaccessible part of the boundary satisfies a (conformal)
flatness condition in one direction, we show that this problem reduces to the
invertibility of a broken geodesic ray transform. In Euclidean space, sets
satisfying the flatness condition include parts of cylindrical sets, conical
sets, and surfaces of revolution. We prove local uniqueness in the Calderon
problem with partial data in admissible geometries, and global uniqueness under
an additional concavity assumption. This work unifies two earlier approaches to
this problem (\cite{KSU} and \cite{I}) and extends both. The proofs are based
on improved Carleman estimates with boundary terms, complex geometrical optics
solutions involving reflected Gaussian beam quasimodes, and invertibility of
(broken) geodesic ray transforms. This last topic raises questions of
independent interest in integral geometry.
