Gelfand-Zetlin polytopes are important in the finite dimensional representation theory of SLn(C) and the symplectic geometry of coadjoint orbits of the unitary group. We examine the combinatorics of Gelfand-Zetlin polytopes in relation to the geometry of the flag variety of SLn(C). The two main contributions of the thesis are as follows: (1) we describe virtual Gelfand-Zetlin polytopes associated to non-dominant weights and (2) we identify the cohomology ring of the flag variety with a quotient of the subalgebra of the Chow cohomology ring of the Gelfand-Zetlin toric variety generated in degree one. More precisely, we take the largest quotient of this subalgebra that satisfies Poincare duality
