A theory for heterogeneous flame propagation in the self-propagating, high-temperature synthesis (SHS) process that proceeds in multi-layer foils consisting of alternating layers of constituents has been formulated, describing a pre-mixed mode of bulk flame propagation supported by a non-pre-mixed reaction that proceeds at the layer surface of a constituent with higher melting point. The formulation allows for volumetric heat loss throughout the bulk flame, a finite-rate Arrhenius reaction at the layer surface, and temperature-sensitive, Arrhenius mass diffusion in the liquid phase. Results show that the burning velocity is inversely proportional to the layer thickness of the constituent with higher melting point; that the burning velocity decreases with increasing heat loss; that at a critical heat-loss rate, the SHS flame extinguishes, as indicated by the characteristic turning-point behavior; that extinction is sensitively affected by the mixture ratio, initial temperature, layer thickness of the constituent with higher melting point, etc.; that the surface reaction is partly or mainly reaction-controlled when the layer is a few nano-meters thick; that effects of the inter-mixed region produced by inter-facial mixing during deposition can be examined by regarding its components as reaction products that play the role of a diluent; and that the critical layer-thickness that the SHS flame ceases to propagate corresponds to the limit of flammability with respect to dilution. It is further shown that the analytical results agree with available experimental data in the literature, indicating that the present formulation captures the essential features of the non-adiabatic, heterogeneous SHS process that self-propagates in multi-layer foils. (C) 2003 The Combustion Institute. All rights reserved.