Solid oxide fuel cells (SOFC) are promising, environmentally friendly energy sources. Many works are devoted to the study of materials, individual aspects of SOFC operation, and the development of devices based on them. However, there is no work covering the entire spectrum of SOFC concepts and designs. In the present review, an attempt is made to collect and structure all types of SOFC that exist today. Structural features of each type of SOFC have been described, and their advantages and disadvantages have been identified. A comparison of the designs showed that among the well-studied dual-chamber SOFC with oxygen-ion conducting electrolyte, the anode-supported design is the most suitable for operation at temperatures below 800 °C. Other SOFC types that are promising for low-temperature operation are SOFC with proton-conducting electrolyte and electrolyte-free fuel cells. However, these recently developed technologies are still far from commercialization and require further research and development.