  We examine domino tilings of rectangular boards, which are in natural
bijection with perfect matchings of grid graphs. This leads to the study of
their associated perfect matching polytopes, and we present some of their
properties, in particular, when these polytopes are Gorenstein. We also
introduce the notion of domino stackings and present some results and several
open questions. Our techniques use results from graph theory, polyhedral
geometry, and enumerative combinatorics.
