A singular perturbation theory is developed for calculating the effects of finite-rate chemical kinetics on the regression rate of a semi-infinite flat plate of fuel placed in a gaseous oxidizer stream. The problem inherently is a coordinate expansion problem, and it is treated as such. Techniques analogous to those applied in matched asymptotic parameter expansions are derived for this particular problem. Descriptions are obtained of the asymptotic solution in which a flame of infinitesimal thickness is present in the boundary layer, of the asymptotic structure of this thin flame, of the diffusion, convection and reaction of oxidizer which has leaked through the flame into the fuel zone due to the finite reaction rate, and of the perturbations to the velocity field and to the linear regression rate of the fuel plate which are produced by the presence of this oxidizer at the wall. It is shown that if the reaction order with respect to the oxidizer exceeds unity, then the parametric dependence of the wall ...