There is considerable technological interest in developingnew fuel-flexible combustion systems that can burn fuels such ashydrogenor syngas. Lean premixed systems have the potential to burn thesetypes of fuels with high efficiency and low NOx emissions due to reducedburnt gas temperatures. Although traditional scientific approaches basedon theory and laboratory experiment have played essential roles indeveloping our current understanding of premixed combustion, they areunable to meet the challenges of designing fuel-flexible lean premixedcombustion devices. Computation, with itsability to deal with complexityand its unlimited access to data, hasthe potential for addressing thesechallenges. Realizing this potential requires the ability to perform highfidelity simulations of turbulent lean premixed flames under realisticconditions. In this paper, we examine the specialized mathematicalstructure of these combustion problems and discuss simulation approachesthat exploit this structure. Using these ideas we can dramatically reducecomputational cost, making it possible to perform high-fidelitysimulations of realistic flames. We illustrate this methodology byconsidering ultra-lean hydrogen flames and discuss how this type ofsimulation is changing the way researchers study combustion.