There has been a recent surge of interest in the combustion of hydrocarbon fuels within porous inert media. The interest has been directed by the needs of industry to develop high performance radiant heaters while complying with increasingly stringent emissions regulations. This paper reviews the processes associated with non-catalytic combustion within porous media, and describes related experimental and modeling research. Experimental measurements in porous media are difficult because of the physical limitations caused for both optical and mechanical probes by the presence of a solid matrix. Modeling is challenging because of the limited knowledge of the fundamentals of the thermal, radiative, and fluid mechanical processes within porous media, and how these participate in the combustion process. Much of the recent interest in the field has followed the commercial availability of reticulated ceramic foams. This paper describes the structural properties of these materials, and their heat transfer properties including convective, conduction, and radiative behavior. The fluid mechanics of porous materials, relating to pressure drop and turbulence flow characteristics are presented. Flames stabilized within the matrix of a porous media have higher burning speeds and leaner flammability limits than open flames. This is due to the internal feedback of heat from the burned gases to the unburned gases through radiation and conduction through the porous medium. This ability to operate at very lean equivalence ratios also contributes to their characteristically low NO emissions. In this paper, we present experimental measurements which show the effects of the porous matrix on reaction rates, flammability limits, and flame stabilization. Exhaust emissions and radiant output from porous media burners are presented for both single-stage and multi-stage burners. The use of liquid fuels in a non-premixed mode of combustion is also discussed. Modeling the combustion process within porous media is quite complex because it requires coupled solution of the energy transfer and chemical kinetics occurring locally in the medium. We present approaches of varying sophistication which predict flame speeds, temperature and concentration profiles,and radiative efficiency.