An overview is given which describes the role of computational modeling in combustion research. The modeling effort is divided into two broad subgroups-fundamental processes and systems simulation-which are amenable to computational solution. The study of fundamental processes is addressed primarily at understanding complex chemical kinetic interactions while systems simulation seeks primarily to quantify complicated hydrodynamic behavior. Even though the two approaches seek to predict different aspects of combustion phenomena, they share a number of common computational difficulties. Four such areas are identified and the scope of the mathematical problems are outlined. These topics are operator-splitting, stiff differential equations, coordinate systems and resolution and iterative minimization techniques.