The transient two-dimensional heterogeneous combustion of fuel on a catalytic flat plate is studied. The solutions are obtained for the three different wall boundary conditions of an isothermal, a perfectly conducting plate, and an adiabatic plate. The laminar boundary layer equations are transformed, and the resulting equations are solved numerically. Equations are nondimensionalized to make the results as general as possible. For a perfectly conducting plate, a universal approach is used which shows the solution of the transport equations to be single valued and that the multiple solutions are caused only by the form of the rate expression. General solutions are found for an isothermal plate which are applicable to any first order reaction or a fuel lean mixture with second order reaction. Solutions for an adiabatic plate show a high temperature region is formed down the plate which moves up. Ignition delay time and ignition distance are found to have the same dependence on the flow and kinetic parameters, except that the ignition delay time does not depend on the free-stream velocity. The results are obtained for CO as well as H~ oxidations.