The volumes and integrals of over polyhedra are perhaps the most fundamental and basic concept in the history of mathematics. Already ancient civilizations worried about it (e.g., Egypt, Babylon) and we teach formulas for volumes of pyramids and cubes to K-6 students. Yet, volumes and integrals of convex polytopes are quite useful still today, from algebraic geometry to computer graphics, from combinatorics to probability and statistics.But, how does one go about actually computing an integral over a convex polytope if one cares to compute the number exactly? In this talk we survey why exact integral computation is relevant, why calculus techniques fail miserably for the goal of computation, and end with the latest results about efficient computation of integrals of polynomials over convex polytopes. If time allows I will demonstrate our new Software LattE Integrale.New results are joint work with: V. Baldoni, N. Berline, B. Dutra, M. Koeppe, M. Vergne.