We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs is to establish conditions that ensure that a marginally outer trapped surface is “weakly outermost”. A rigidity result for Riemannian manifolds with a lower bound on their scalar curvature is included as a special case. Relevant background on marginally outer trapped surfaces will be discussed. This talk is based on joint work with Michael Eichmair and Abraão Mendes.