Gaseous detonation is a phenomenon with very complicated dynamics whichhas been studied extensively by physicists, mathematicians and engineers for many years.Despite many efforts the problem is far from a complete resolution. Recently the theoryof subsonic detonation that occurs in highly resistant porous media was proposed in [4].This theory provides a model which is realistic, rich and suitable for a mathematical study.In particular, the model is capable of describing the transition from a slowly propagatingdeflagration wave to the fast detonation wave. This phenomena is known as a deflagrationto detonation transition and is one of the most challenging issues in combustion theory. Inthis paper we will present some recent mathematical results concerning initiation of reactionin porous media, existence and uniqueness of traveling fronts, quenching and propagation.