A variety of advanced materials can be synthesized using solid-state combustion reactions, following two different modes of synthesis, the self-propagating mode and the thermal explosion mode. Combustion synthesis offers many potential advantages over conventional techniques of synthesis, including relatively simple equipment, shorter processing times, lower energy requirements, higher product purities, and the possibility to synthesize metastable phases. Various aspects of combustion synthesis are discussed in this review. They include thermodynamic considerations, theory and mechanisms, and structure-forming processes. The influence of processing variables on reaction progress and structure formation is discussed in detail. Some novel technologies and materials based on combustion synthesis are also described. The principles of chemical reaction engineering, combined with concepts and techniques of materials science, provide a suitable basis for studies of combustion synthesis.