  It is well-known that considerations of symmetry lead to the definition of a
host of conserved quantities (energy, linear momentum, center of mass, etc.)
for an asymptotically flat initial data set, and a great deal of progress in
Mathematical Relativity in recent decades essentially amounts to establishing
fundamental properties for such quantities (positive mass theorems, Penrose
inequalities, geometric representation of the center of mass by means of
isoperimetric foliations at infinity, etc.) under suitable energy conditions.
In this article I first review certain aspects of this classical theory and
then describe how they can be (partially) extended to the setting in which the
initial data set carries a non-compact boundary. In this case, lower bounds for
the scalar curvature in the interior and for the mean curvature along the
boundary both play a key role. Our presentation aims to highlight various
rigidity/flexibility phenomena coming from the validity, or lack thereof, of
the corresponding positive mass theorems and/or Penrose inequalities.
