The structure of one-dimensional flames is shown to be completely determined by constants such as those of heat conductivity, of diffusion, and of the homogeneous reaction rates. The mathematical problem in the most general case is intractable, but three simple cases are solved fully by a mathematical method of successive approximations. The three cases are those in which diffusion is neglected compared with heat conductivity and in which reaction velocities of the following types are considered: unimolecular, bimolecular, and the quasibimolecular form of a unimolecular reaction at low pressures. The method of mathematical approximation is shown to involve errors of the order of only 10% in some actual cases, an error which is negligible compared with other uncertainties of the problem. In these simple cases it is possible to solve all details of the structure such as the variation of composition and temperature through the flame.