||We use generating functions and complex-analytic methods to count integer lattice points in polytopes with rational vertices. More precisely, we study the number of lattice points as the polytope gets dilated by an integer factor. This expression is known as the Ehrhart quasipolynomial. Because polytopes can be described by a system of linear equalities and inequalities, they appear in a wealth of areas. We will show applications of Ehrhart quasipolynomials to number theory, combinatorics, and computational geometry, illustrating (or so we hope) that pure mathematics and computationally efficient algorithms are not mutually exclusive.