DMDO is most commonly used for the oxidation of alkenes to epoxides.  One advantage of using DMDO is that the only byproduct of oxidation is acetone, a fairly innocuous and volatile compound. DMDO oxidations are particularly mild, sometimes allowing oxidations which might not otherwise be possible. 
Despite its high reactivity, DMDO displays good selectivity for electron-rich olefins.  DMDO will also oxidize several other functional groups.  For example, DMDO will oxidize primary amines to nitro compounds and sulfides to sulfoxides.  In some cases, DMDO will even oxidize unactivated C-H bonds:
DMDO can also be used to convert nitro compounds to carbonyl compounds (Nef reaction).