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Given a partition $\lambda$ and a composition $\mu$ of a positive integer $n$, the Gelfand--Tsetlin polytope $GT(\lambda, \mu) \subset \R^{n(n+1)/2}$ is a convex rational polytope with the property that the number of integral lattice points in $GT(\lambda, \mu)$ equals the dimension of the weight $\mu$ subspace of the irreducible representation of the Lie algebra $\mathfrak{gl}_n \C$ with highest weight $\lambda$. We discuss the Ehrhart quasi-polynomials of these polytopes and introduce an elementary combinatorial construction, called a tiling, for determining the denominators that can appear in the coordinates of their vertices. These tilings allow the construction of counterexamples to a conjecture of Berenstein and Kirillov (1995) that the vertices of GT-polytopes are always integral. Details appear in a joint paper with Prof. De Loera, which is available on the ArXiv at http://front.math.ucdavis.edu/math.CO/0309329.