Interstellar dust grains play a crucial role in the evolution of the galactic interstellar medium (ISM). Despite its importance, however, dust remains poorly understood in terms of its origin, composition, and abundance throughout the universe. Supernova remnants (SNRs) provide a laboratory for studying the evolution of dust grains, as they are one of the only environments in the universe where it is possible to observe grains being both created and destroyed. SNRs exhibit collisionally heated dust, allowing dust to serve as a diagnostic both for grain physics and for the plasma conditions in the SNR. I present theoretical models of collisionally heated dust which calculate grain emission as well as destruction rates. In these models, I incorporate physics such as nonthermal sputtering caused by grain motions through the gas, a more realistic approach to sputtering for small grains, and arbitrary grain compositions porous and composite grains. I apply these models to infrared and X-ray observations of Kepler's supernova and the Cygnus Loop in the galaxy, and SNRs 0509-67.5, 0519-69.0, and 0540-69.3 in the LMC. X-ray observations characterize the hot plasma while IR observations constrain grain properties and destruction rates. Such a multi-wavelength approach is crucial for a complete understanding of gas and dust interaction and evolution. Modeling of both X-ray and IR spectra allows disentangling of parameters such as pre and postshock gas density, as well as swept-up masses of gas and dust, and can provide constraints on the shock compression ratio. Observations also show that the dust-to-gas mass ratio in the ISM is lower by a factor of several than what is inferred by extinction studies of starlight. Future observatories, such as the James Webb Space Telescope and the International X-ray Observatory, will allow testing of models far beyond what is possible now.