The heating of circumstellar and interstellar dust grains of various types by extragalactic supernovae is investigated. Models to compute the infra-red flux from such dust grains are devised. The optical constants of different grain materials are deduced, and the method to compute the temperature of a grain heated by a supernova is detailed. The evaporation and condensation of grains are considered and the effect of forces on the grains are investigated. Spectra and light curves are computed for radiation from interstellar graphite and silicate grains in a typical nearby galaxy heated by a type II supernova. Spectra of circumstellar silicate dust shells around the supernovae SN1979c and SN1980k are computed and compared with observational data. This comparison leads to a determination of the masses and sizes of the circumstellar dust shells which condense around these supernovae. Finally the formation of a composite double dust shell consisting of an inner shell of homogeneous grains and an outer shell of core-mantle grains is considered. The condensation of such a dust shell around a type II supernova is found to be possible. Spectra are computed for this type of circumstellar dust shell and compared with the spectra computed for a circumstellar dust shell of homogeneous silicate grains.