This paper is a study of the polyhedral geometry of Gelfand–Tsetlin polytopes arising in the representation theory of \frak gl_n \Bbb C{\frak gl}_n \Bbb C and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand–Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand–Tsetlin polytopes. We can construct for each n ³ 5n\geq5 a counterexample, with arbitrarily increasing denominators as nn grows, of a nonintegral vertex. This is the first infinite family of nonintegral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the nonintegral vertices when nn is fixed.