Brion's formula is an elegant formula for computing the integral of an exponential function exp(<alpha, x >) over a polytope P; there is also a version for computing the sum of an exponential function over the lattice points in P. A naive version of these formulas breaks down if alpha is orthogonal to any one-dimensional face of P. In the first half of this talk we will discuss these formulas, as well as a "degenerate" version in case alpha is in fact orthogonal to some face. In the second half we will discuss how these formulas are useful for analysis on symmetric spaces (in the continuous case) and on affine buildings (in the discrete case), focusing in particular on volume growth entropy.