There is considerable technological interest in developing new fuel-flexible combustion systems that can burn fuels such as hydrogenor syngas. Lean premixed systems have the potential to burn these types of fuels with high efficiency and low NOx emissions due to reduced burnt gas temperatures. Although traditional scientific approaches based on theory and laboratory experiment have played essential roles in developing our current understanding of premixed combustion, they are unable to meet the challenges of designing fuel-flexible lean premixed combustion devices. Computation, with itsability to deal with complexity and its unlimited access to data, hasthe potential for addressing these challenges. Realizing this potential requires the ability to perform high fidelity simulations of turbulent lean premixed flames under realistic conditions. In this paper, we examine the specialized mathematical structure of these combustion problems and discuss simulation approaches that exploit this structure. Using these ideas we can dramatically reduce computational cost, making it possible to perform high-fidelity simulations of realistic flames. We illustrate this methodology by considering ultra-lean hydrogen flames and discuss how this type of simulation is changing the way researchers study combustion.