  In this expository article we give an introduction to Ehrhart theory, i.e.,
the theory of integer points in polyhedra, and take a tour through its
applications in enumerative combinatorics. Topics include geometric modeling in
combinatorics, Ehrhart's method for proving that a couting function is a
polynomial, the connection between polyhedral cones, rational functions and
quasisymmetric functions, methods for bounding coefficients, combinatorial
reciprocity theorems, algorithms for counting integer points in polyhedra and
computing rational function representations, as well as visualizations of the
greatest common divisor and the Euclidean algorithm.
