Many inflating spacetimes are likely to violate the weak energy condition, a key assumption of many singularity theorems. In this talk I will describe a recent theorem by Vilenkin, Borde, and me which uses a simple kinematical argument, independent of any energy conditions, to show that a cosmological model that is inflating-- or just expanding sufficiently fast-- must be incomplete in null and timelike past directions. Specifically, we obtained a bound on the integral of the Hubble parameter over a past-directed timelike or null geodesic. Thus inflationary models require physics other than inflation to describe the past boundary of the inflating region of spacetime.