Abstract: Singular perturbation is widely used to obtain the approximate solutions for mechanical problems. A typical problem is the boundary layer in fluid mechanics. Yung-Huai Kuo has developed the singular perturbation theory for the boundary layer over a plate. Similar to the boundary layer in fluid mechanics, the laminar premixed flame in combustion can also be analyzed by the singular perturbation method, which is usually called as the large-activation-energy asymptotics. The laminar premixed flame structure consists of the preheat zone, reaction zone, and equilibrium zone. Under the limit of the large activation energy, the chemical reaction rate is very sensitive to the temperature and thereby chemical reaction only occurs in the very thin reaction zone. The ratio between the reaction zone thickness and the preheat zone thickness is a small parameter, based on which the asymptotic analysis can be conducted for a laminar premixed flame. This paper reviews the application and progress of the large-activation-energy asymptotic analysis in the one-dimensional planar premixed flame and spherically propagating flame. First, the structure of the premixed flame and its different characteristic length scales are introduced. The length scale separation due to large activation energy is analyzed. The detailed derivation of the large-activation-energy asymptotic analysis of a planar premixed flame is presented. The analytical solutions for the preheat zone, reaction zone, and equilibrium zone are first sepratedly obtained and then matched aound the interfaces among these three zones. The effects of radiation heat loss on premixed planar flames are discussed. Then, the application of the large-activation-energy asymptotic analysis to the ignition and spherically propagating flame is introduced. It is pointed out that in order to accurately predict the critical ignition conditions, the theory should be able to describe both the ignition kernel development and the spherical flame propagation afterwards. The ignition and flame propagation theory considering chain reactions is discussed, and the trend of theoretical research on flame instability is described. Moreover, the effects of radiation heat loss on spherical flame propagation are discussed. Finally, the future research directions are prospected based on the current research progress, which includes multi-step chemistry, low-temperature cool flame, complicated flow and radiation reabsorption.