This review considers a specific domain in combustion science, so-called discrete combustion waves, which have been observed in a variety of processes including combustion synthesis of materials, burning of solid propellants, coals and biomass. The main objective of this work is to discuss the relevant experimental and theoretical results and to bring to light the fundamental nature of this phenomenon. The criteria for distinguishing between the homogeneous and discrete combustion waves based on the analysis of local and global behavior of the reaction systems have been discussed. Different theoretical models that account the discrete nature of the process are critically overviewed and compared with experimental results. It was concluded that combustion occurs in quasi-homogeneous or discrete modes depending on the ratio of characteristic times of chemical reaction and heat transfer in the heterogeneous medium. (C) 2007 Elsevier Ltd. All rights reserved.