Heat transfer in regenerators when the hot and the cold gases flow alternatively and in crossflow directions through the material space is investigated. Exact analytical solutions for the temperatures of gases and solid material as function of time and space are presented for the general asymmetric and unbalanced case. The solutions can readily be used to dimension the regenerator operating in the crossflow mode. In cyclic stationary regenerators the gas outlet temperature varies with time and the damping of the temperature fluctuations is discussed. Applications in combustion of fuels with low heating value are presented.