  This article deals with stability issues related to geodesic X-ray
transforms, where an interplay between the (attenuation type) weight in the
transform and the underlying geometry strongly impact whether the problem is
stable or unstable. In the unstable case, we also explain what types of
artifacts are expected in terms of the underlying conjugate points and the
microlocal weights at those points. We show in particular that the well-known
iterative reconstruction Landweber algorithm cannot provide accurate
reconstruction when the problem is unstable, though the artifacts generated,
specific for the reconstruction algorithm, can be properly described.
