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.