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The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. I will introduce this topic and discuss a theorem stating that for fixed $d$, there exists a bounded region of the complex plane containing all roots of Ehrhart polynomials of d-polytopes, and that all real roots of these polynomials lie in $[-d, d/2)$. In contrast, we prove that when the dimension $d$ is not fixed the positive real roots can be arbitrarily large. This is joint work with M. Beck, M. Develin, J. Pfeifle and R. P. Stanley.