The Yamabe invariant of an asymptotically Euclidean (AE) manifold is defined analogously to that of a compact manifold. Nevertheless, the prescribed scalar curvature problem in the AE setting has features that are quite different from its compact counterpart. For example, a Yamabe positive AE manifold admits a conformally related metric that has a scalar curvature with any desired sign: positive, negative or zero everywhere. In this talk we discuss the resolution of the prescribed nonpositive scalar curvature problem for AE manifolds and its application to general relativity.