  We perform a systematic analysis of dark matter production during
post-inflationary reheating. Following the period of exponential expansion, the
inflaton begins a period of damped oscillations as it decays. These
oscillations and the evolution of temperature of the thermalized decay products
depend on the shape of the inflaton potential $V(\Phi)$. We consider potentials
of the form $\Phi^k$. Standard matter-dominated oscillations occur for $k=2$.
In general, the production of dark matter may depend on either (or both) the
maximum temperature after inflation, or the reheating temperature, where the
latter is defined when the Universe becomes radiation dominated. We show that
dark matter production is sensitive to the inflaton potential and depends
heavily on the maximum temperature when $k>2$. We also consider the production
of dark matter with masses larger than the reheating temperature.
