We describe a simple, fast algorithm for evaluating a Volterra convolution integral operator at N points of a grid with a cost that is almost linear in N. By comparison, the cost is quadratic in N using the obvious approach. The fast algorithm requires the kernel k(t) to be approximated by a sum of terms of the form w exp(-at) for t in a compact interval bounded away from zero. This type of nonlinear approximation has proved useful for implementing nonreflecting boundary conditions for wave equations. I will describe a more recent application to fractional PDEs, and a new method for deriving the necessary exponential sum approximation.