This chapter develops the geometry of and analysis on initial data sets that arise in models of isolated gravitational systems. We begin with some detailed discussion and analysis involving the Laplace operator on asymptotically flat manifolds, which we use to develop density and deformation results on scalar curvature, leading to a proof of the Riemannian positive mass theorem. In the last section of the chapter we develop a technique for localized scalar curvature deformation, and we apply it to glue an asymptotically flat end with vanishing scalar curvature to an end of a Riemannian Schwarzschild metric, maintaining zero scalar curvature throughout.