An Anosov manifold with boundary is a compact smooth Riemannian manifold M with strictly convex boundary, hyperbolic trapped set (possibly empty), and no conjugate points. We will discuss when M admits a codimension zero isometric embedding into a closed Riemannian manifold with Anosov geodesic flow and demonstrate how to use such embeddings to prove the marked boundary distance rigidity conjecture for Anosov surfaces with boundary. This talk is based on joint work with Dong Chen and Andrey Gogolev as well as joint work with Thibault Lefeuvre.