Combustion (self propagating high temperature) synthesis (SHS) reactions have been classified into (i) simple reaction systems in which no other processing step, other than the SHS reaction, is considered; (ii) SHS reactions which involve simultaneous and/or subsequent densification process(es); (iii)SHS reactions coupled with vapor transport systems and those which synthesize specific products such as functionally-graded materials (FGM), thin films and coatings, and mechanically alloyed powders or those used to join dissimilar ceramics and/or intermetallics. Finally, a comparison is made of the various mathematical models that have been developed to simulate and predict instabilities in propagating combustion synthesis reactions.