Heat and electricity generating apparatus (201) for a boiler system (202) for use in a residential or small-scale building. The heat and electricity generating 5 apparatus (201) comprises a centrifugal air compressor (203), a turbo-expander (204) and an electric generator (205) operatively connected by a rotatable drive shaft (206) and further comprises a high-pressure combustor (207), a first air-to- air heat exchanger (208), a second air-to-fluid heat exchanger (209) and a diverter gate (222). In operation, a method of heat transfer is performed in which combustion product gas issued by the high-pressure combustor (207) is passed to the first heat exchanger (208) either through the turbo-expander (204) or directly, to heat compressed air issued by the centrifugal air compressor (203) prior to entry into the high-pressure combustor (207), and is then passed through the second heat exchanger (209), to heat a heat-conveying fluid.