In this talk I discuss recent joint research with Pilar Herreros on a rigidity result for Riemannian manifolds with boundary. Our main result is that if such a manifold is a priori known to be real-analytic and also known to satisfy some mild conditions, then it is uniquely determined up to isometry by its scattering data. Informally, the scattering data can be described as a map that takes a point and direction of entry of a geodesic ray and returns the point and direction of exit. The general problem of scattering rigidity is a natural generalization of the well-studied problem of boundary distance rigidity for Riemannian manifolds. Our main result improves on earlier theorems in that it does not require any geodesic length data and also has very few restrictions on conjugate points. However, we require the admittedly heavy restriction that everything be real-analytic.