Single-photon sources are one of the key components in quantum photonics applications. These sources ideally emit a single photon at a time, are highly efficient, and could be integrated in photonic circuits for complex quantum system designs. Various platforms to realize such sources have been actively studied, among which semiconductor quantum dots have been found to be particularly attractive. Furthermore, quantum dots embedded in bottom-up-grown III-V compound semiconductor nanowires have been found to exhibit relatively high performance as well as beneficial flexibility in fabrication and integration. Here, we review fabrication and performance of these nanowire-based quantum sources and compare them to quantum dots in top-down-fabricated designs. The state of the art in single-photon sources with quantum dots in nanowires is discussed. We also present current challenges and possible future research directions.