The dependences of the temperature and the rate of filtration combustion (FC) of carbon on the main parameters controlling the process were experimentally and theoretically studied. Among these are the fraction of the combustible in the charge, the flow rate of the oxidant, the reactivity of the carbon material, and the level of heat loss via the reactor walls. The transition from the conditions of stationary propagation of the combustion wave to decay occurs critically, upon a minor change in the controlling parameters. The parametric domain of the stationary wave was determined. A simplified unidimensional one-temperature mathematical model was proposed for the description of stationary regimes of FC of solid fuels. The model qualitatively characterizes the dependence of the combustion regime on the main controlling parameters and makes it possible to reveal the critical conditions for the existence of stationary combustion regimes.