  We study the boundary and lens rigidity problems on domains without assuming
the convexity of the boundary. We show that such rigidities hold when the
domain is a simply connected compact Riemannian surface without conjugate
points. For the more general class of non-trapping compact Riemannian surfaces
with no conjugate points, we show lens rigidity. We also prove the injectivity
of the X-ray transform on tensors in a variety of settings with non-convex
boundary and, in some situations, allowing a non-empty trapped set.
