This review updates information on the state of development, properties, materials and possible applications of two basic types of high-temperature fuel cells: solid oxide fuel cells (SOFC) and molten carbonate fuel cells (MCFC). The main fuels for such cells are still hydrogen, carbon monoxide and their mixtures, which can be produced from natural gas by steam reforming. A new potential liquid fuel for such cells is in particular methanol. Hydrocarbon fuels have to be reformed with steam prior to utilisation in high-temperature fuel cells. Solid fuels, such as coal and wood, can be practically used as fuels for the cells only after gasification and efficient cleaning of the gaseous fuel. The optimum temperature for MCFC is about 650 ^oC, for SOFC usually between 750 and 900 ^oC, depending on materials. The theoretical thermodynamic maximum efficiencies of the fuel cells for conversion of chemical energy to useful work is high, exceeding 66 %. The real behaviour and efficiency depend on internal losses, construction and current load of the cells. The highest attainable current densities in electric circuits with MCFC and SOFC are still relatively low, below 800 - 1000 mA.cm^-2. Great potential of high-temperature fuel cells is in utilization of waste heat of the produced gases in steam boiler, gas turbine (for pressurized systems) and in cogeneration of electrical energy and heat. Overall real attainable efficiency of the heat-to-power conversion estimated for such combined systems exceeds 60 %.