Debris disks around other stars, like the disks around Fomalhaut, Vega, and Epsilon Eridani, are often described as more massive versions of the Kuiper Belt. But for a long time, it's been hard to test this notion, because grain-grain collisions dominate the grain lifetimes and we lacked the tools to model the effect of collisions on the appearance of the disks. I'll describe a new breakthrough that has allowed us to make 3-D models of collisions in debris disks and exozodiacal clouds for the first time, and I'll show the latest supercomputer simulations of these systems, illustrating the effects of planets and collisions in sculpting these disks. These models will be the key to interpreting debris disk images from HST, Herschel, SOFIA, JWST, and ALMA, as well as understanding the exozodiacal dust backgrounds for direct imaging of exo-Earths
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