Solar-energy conversion usually takes one of two forms: the `quantum' approach, which uses the large per-photon energy of solar radiation to excite electrons, as in photovoltaic cells, or the `thermal' approach, which uses concentrated sunlight as a thermal-energy source to indirectly produce electricity using a heat engine. Here we present a new concept for solar electricity generation, photon-enhanced thermionic emission, which combines quantum and thermal mechanisms into a single physical process. The device is based on thermionic emission of photoexcited electrons from a semiconductor cathode at high temperature. Temperature-dependent photoemission-yield measurements from GaN show strong evidence for photon-enhanced thermionic emission, and calculated efficiencies for idealized devices can exceed the theoretical limits of single-junction photovoltaic cells. The proposed solar converter would operate at temperatures exceeding 200°C, enabling its waste heat to be used to power a secondary thermal engine, boosting theoretical combined conversion efficiencies above 50%.