Impact cratering is the most common geologic process in the solar system. The morphology of the crater produced by the impact of an asteroid or comet onto a planetary surface is a complex function of the kinetic energy of the impactor (velocity and mass), the gravity of the surface being impacted, and the material properties of the target and impactor. Because of the dependence on target morphology, craters are useful indicators of subsurface structure. However, to use craters as target probes we must first understand how target properties change the cratering process and the final crater shape. Thus the goal of this thesis is to use numerical modeling to understand how target properties, such as strength layering and the presence of ice, modify cratering. We compare the results with observations of craters on the moon, Mars, and the icy satellites and explain some of the unique attributes of craters on these bodies. The final chapter addresses the question of the mechanism that leads to extensive weakening during the collapse of large craters.