The performance of an irreversible Brayton heat engine is analysed. Analytical formulae related to power output, pressure ratio and efficiency of the heat engine coupled to either constant- or variable-temperature heat reservoirs are derived. The irreversibilities considered include heat-resistance losses in the hot- and cold-side heat-exchangers, and non-isentropic expansion and compression losses in the turbine and compressor. The optimal performance characteristics of the heat engine are obtained by optimisation of the distribution of heat conductances or heat-transfer surface areas between the heat-exchangers, and the matching of working fluid and heat reservoirs.