We enunciate the general principles that govern the transport of charge and heat in a thermionic device. We illustrate the application of these principles to the subject of domestic refrigeration. A complementary application is power generation. We distinguish Class 2 devices, in which the potential barrier on the hot side plays a role, from Class 1 devices, in which this barrier is irrelevant. We show that the effect of heat backflow is to drastically reduce the efficiency of thermionic devices in both GaAs and InSb representative semiconductor systems. We conclude that practical devices are not likely with bulk, single-barrier devices.