Single photons are a key, fundamental element of most quantum optical technologies, be it for the development of large-scale quantum communication networks, for quantum simulation, or for connecting quantum memories in a quantum computer. The ideal single-photon source is an on-demand, deterministic single-photon source: it delivers light pulses in a well-defined polarization and spatiotemporal mode and containing exactly one single photon. For many applications, there is a quantum advantage only if the single photons are highly indistinguishable in all their degrees of freedom. So far, many quantum optical technologies have used single-photon sources based on parametric down- conversion. While excellent for small demonstrations, they cannot be used effectively in large-scale quantum optical systems due to their intrinsic multiphoton nature and low brightness when scaled. In 2001, semiconductor quantum dots were shown to emit single photons, thereby opening a path to integrated single-photon sources. However, it has proven scientifically and technologically challenging to engineer such sources to have high brightness and photon indistinguishability. We discuss here these challenges and review the spectacular progresses that have been achieved in the last few years. The latest quantum dot-based single-photon sources are edging closer to the ideal single- photon source, and have opened new possibilities for quantum technologies.