Abstract: An Einstein manifold is called scalar curvature rigid if there are no compactly supported volume-preserving deformations of the metric which increase the scalar curvature. We give various characterizations of scalar curvature rigidity for closed as well as for open Einstein manifolds which are related to spectral properties of the Lichnerowicz Laplacian. For asymptotically hyperbolic manifolds, we construct a new mass invariant, called the volume-renormalized mass, which we use to characterize scalar curvature rigidity and stability of Poincaré-Einstein manifolds. This is joint work with Mattias Dahl and Stephen McCormick.