  In many models of inflation, the period of accelerated expansion ends with
preheating, a highly non-thermal phase of evolution during which the inflaton
pumps energy into a specific set of momentum modes of field(s) to which it is
coupled. This necessarily induces large, transient density inhomogeneities
which can source a significant spectrum of gravitational waves. In this paper,
we consider the generic properties of gravitational waves produced during
preheating, perform detailed calculations of the spectrum for several specific
inflationary models, and identify problems that require further study. In
particular, we argue that if these gravitational waves exist they will
necessarily fall within the frequency range that is feasible for direct
detection experiments -- from laboratory through to solar system scales. We
extract the gravitational wave spectrum from numerical simulations of
preheating after $\lambda \phi^4$ and $m_{\phi}^2 \phi^2$ inflation, and find
that they lead to a gravitational wave amplitude of around $\Omega_{gw}h^2\sim
10^{-10}$. This is considerably higher than the amplitude of the primordial
gravitational waves produced during inflation. However, the typical wavelength
of these gravitational waves is considerably shorter than LIGO scales, although
in extreme cases they may be visible at scales accessible to the proposed BBO
mission. We survey possible experimental approaches to detecting any
gravitational wave background generated during preheating.
