Many inflating spacetimes are likely to violate the weak energy condition, a key assumption of singularity theorems. In this talk I will describe work with Borde and Vilenkin in which we developed a simple kinematical argument, without assuming any energy condition, 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 some physics other than inflation -- for example, a mechanism of quantum creation -- to describe the past boundary of the inflating region of spacetime.