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.