The gravitational-wave observatories LIGO and Virgo have so far detected several tens of merging compact binaries, allowing for the first time to study their population properties and formation scenarios, while many more detections are expected with upcoming observations. However, many astrophysical sources are either very faint or produce signals that overlap in time, and thus cannot be detected individually. The superposition of these independent and unresolved sources creates a stochastic background. In this talk I will discuss the various astrophysical gravitational-wave backgrounds and their detection prospects with ground-based experiments, pulsar timing arrays and the space-based LISA mission. I will specifically focus on stellar-mass and super-massive black hole binaries and the complementarity between stochastic backgrounds and individual detections in studying black hole populations.