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I’ll describe some recent results on the geometry and topology of manifolds with positive scalar curvature (PSC), and the related subject of the constraint equations of general relativity. I’ll cover a number of results which generalize various classics : Schoen-Yau’ 1979 topological obstructions to PSC, Schoen-Yau’s 1979 Positive Mass Theorem, and the Density Theorem of Corvino-Schoen 2005. In dimensions less than 8, these results settle the positive mass with arbitrary ends conjecture made in Schoen-Yau 1988, and the positive mass theorem for initial data sets with boundary (the case without boundary was resolved by Eichmair-Huang-Lee-Schoen 2015, Huang-Lee 2017, Eichmair 2013). Time permitting, I’ll discuss the proof of one of these in more detail. This is based on four papers, two with R.Unger and S.T.Yau, and two with D.Lee and R.Unger.