Discovered in the 1730s, the classical Euler-Maclaurin formula may be viewed as a formula for summing the values of a function over the lattice points in a one-dimensional polytope. Several years ago, Berline and Vergne generalized this formula to polytopes of arbitrary dimension, obtaining a formula for the sum of a polynomial function over the lattice points in any rational polytope. This talk will present an algebraic approach to the Euler-Maclarurin problem for polytopes based on recent results of Fischer and the speaker. While this approach is motivated from the theory of toric varieties, in this talk we aim to tell the story in a way that does not directly use results from toric geometry.