The problem of quenching a material in self-propagating high-temperature synthesis (SHS) in flat samples of a titanium-carbon mixture is studied theoretically and experimentally. Possible cooling rates with different ways of heat removal are estimated mathematically. A method of quenching by a high-velocity impinging jet is chosen. The method provides the conditions of heat removal necessary for stopping the combustion wave front (a rate of temperature decrease similar or equal to 10(4) K s(-1) for a 2-mm-thick specimen). A mathematical model of material quenching in the process of SHS is developed for one- (A --> B) and two-stage (A --> B --> C) reactions. Thermal processes and composition of material in stopping the front in st flat plate sample of a titanium-carbon mixture is studied numerically on the basis of the model proposed. The dependence of the cooling rate on temperature in different zones of an SHS-wave are calculated; in this case a two-stage reaction is considered to occur in the so-called fusion, control, and separation modes. A schematic diagram of an experimental setup and results of experimental study of the proposed model of quenching are given. Feasibility of using the method of quenching by an impinging jet is demonstrated for studying the processes of phase and structure formation during the SHS.