A well known result is that a Riemannian (spin) manifold of non-negative scalar curvature which is sufficiently asymptotically flat has to be the Euclidean space. This scalar curvature rigidity result of the Euclidean space can be generalized to certain symmetric spaces of sectional curvature $K\leq 0$. Various examples will be presented in the talk and moreover, a way to classify these "scalar curvature rigid" symmetric spaces is shown.