||In this talk we will see some basic facts about Ehrhart polynomials after introducing the necessary notions from polyhedral geometry. The Vector Partition Function, which can be thought of as a generalization of Ehrhart polynomials, will be then explored from both a geometric and an analytic perspective. In particular, we will see that the Vector Partition Function is piecewise (quasi-)polynomial. The regions of polynomiality form a complex which we will compute in two different ways, using Partition Analysis and geometry. The geometric way is based on a simple observation about the connection of the polynomiality complex to a certain cone constructed for the computation of the indecomposable Minkowski summands of a polytope. If time permits, returning to the study of Ehrhart polynomials, we will see how to use geometry to interpolate Ehrhart polynomials.