  The collection of individually resolvable gravitational wave (GW) events
makes up a tiny fraction of all GW signals which reach our detectors, while
most lie below the confusion limit and go undetected. Like voices in a crowded
room, the collection of unresolved signals gives rise to a background which is
well-described via stochastic variables, and hence referred to as the
stochastic GW background (SGWB). In this review, we provide an overview of
stochastic GW signals, and characterise them based on features of interest such
as generation processes and observational properties. We then review the
current detection strategies for stochastic backgrounds, offering a
ready-to-use manual for stochastic GW searches in real data. In the process, we
distinguish between interferometric measurements of GWs, either by ground-based
or space-based laser interferometers, and timing-residuals analyses with pulsar
timing arrays (PTAs). These detection methods have been applied to real data
both by the large GW collaborations and smaller research groups, and the most
recent and instructive results are reported here. We close this review with an
outlook on future observations with third generation detectors, space-based
interferometers, and potential non-interferometric detection methods proposed
in the literature.
