Topics in Hyperplane Arrangements, Polytopes and Box-SplinesPreliminaries. Polytopes. Hyperplane Arrangements. Fourier and Laplace Transforms. Modules over the Weyl Algebra. Differential and Difference Equations. Approximation Theory I. The Di? erentiable Case. Splines. RX as a D Module. The Function TX. Cohomology. Differential Equations. The Discrete Case. Integral Points in Polytopes. The Partition Functions. Toric Arrangements. Cohomology of Toric Arrangements. Polar Parts. Approximation Theory.