A general asymptotic formulation is presented for diffusion flames of large-activation-energy chemical reactions. In this limit chemical reaction is confined to a thin zone which, when viewed from the much larger diffusion scale, is a moving two-dimensional sheet. The formulation is not restricted to any particular configuration, and applies to conditions extending from complete combustion to extinction. The detailed structure of the reaction zone yields jump conditions that permit full determination of the combustion field on both sides of the reaction zone, as well as the instantaneous shape of the reaction sheet itself. The simplified system is subsequently used to study the intrinsic stability properties of diffusion flames and, in particular, the onset of cellular flames. We show that cellular diffusion flames form under near-extinction conditions when the reactant in the feed stream is the more completely consumed reactant, and the corresponding reactant Lewis number is below some critical value. Cell sizes at the onset of instability are on the order of the diffusion length. Predicted cell sizes and conditions for instability are therefore both comparable with experimental observations. Finally, we provide stability curves in the fuel and oxidant Lewis number parameter plane, showing where instability is expected for different values of both the initial mixture strength and the Damköhler number.