  We study inverse boundary problems for magnetic Schr\"odinger operators on a
compact Riemannian manifold with boundary of dimension $\ge 3$. In the first
part of the paper we are concerned with the case of admissible geometries, i.e.
compact Riemannian manifolds with boundary which are conformally embedded in a
product of the Euclidean line and a simple manifold. We show that the knowledge
of the Cauchy data on the boundary of the manifold for the magnetic
Schr\"odinger operator with $L^\infty$ magnetic and electric potentials,
determines the magnetic field and electric potential uniquely.
  In the second part of the paper we address the case of more general
conformally transversally anisotropic geometries, i.e. compact Riemannian
manifolds with boundary which are conformally embedded in a product of the
Euclidean line and a compact manifold, which need not be simple. Here, under
the assumption that the geodesic ray transform on the transversal manifold is
injective, we prove that the knowledge of the Cauchy data on the boundary of
the manifold for a magnetic Schr\"odinger operator with continuous potentials,
determines the magnetic field uniquely. Assuming that the electric potential is
known, we show that the Cauchy data determines the magnetic potential up to a
gauge equivalence.
